0, the numerator is approximately equal to the difference between the two areas, which is the area under the graph of f from x to x + h. That is: If we divide both sides of the above approximation by h and allow _h_→0, then: This is always true regardless of whether the f is positive or negative. If f is a continuous function, then the equation above tells us that F(x) is a differentiable function whose derivative is f. This can be represented as follows: In order to understand how this is true, we must examine the way it works. The fundamental theorem of calculus is central to the study of calculus. Using the fundamental theorem of calculus, evaluate the following: In Part 1 of the Fundamental Theorem of Calculus, we discovered a special relationship between differentiation and definite integrals. This is surprising – it’s like saying everyone who behaves like Steve Jobs is Steve Jobs. Ok. Part 1 said that if we have the original function, we can skip the manual computation of the steps. Differentiate to get the pattern of steps. We know the last change (+9) happens at $$x=4$$, so we’ve built up to a 5$$\times$$5 square. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The key insights are: In the upcoming lessons, we’ll work through a few famous calculus rules and applications. This is a very straightforward application of the Second Fundamental Theorem of Calculus. For instance, if we let G(x) be such a function, then: We see that when we take the derivative of F - G, we always get zero. The practical conclusion is integration and differentiation are opposites. The real goal will be to figure out, for ourselves, how to make this happen: By now, we have an idea that the strategy above is possible. Newton and Leibniz utilized the Fundamental Theorem of Calculus and began mathematical advancements that fueled scientific outbreaks for the next 200 years. In my head, I think “The next step in the total accumulation is our current amount! Technically, a function whose derivative is equal to the current steps is called an anti-derivative (One anti-derivative of $$2$$ is $$2x$$; another is $$2x + 10$$). The Fundamental Theorem of Calculus gently reminds us we have a few ways to look at a pattern. But how do we find the original? Makes things easier to measure, I think.”). In all introductory calculus courses, differentiation is taught before integration. Let me explain: A Polynomial looks like this: example of a polynomial this one has 3 terms: Thus, the two parts of the fundamental theorem of calculus say that differentiation and integration are inverse processes. It bridges the concept of … This must mean that F - G is a constant, since the derivative of any constant is always zero. If derivatives and integrals are opposites, we can sidestep the laborious accumulation process found in definite integrals. I hope the strategy clicks for you: avoid manually computing the definite integral by finding the original pattern. First, if you take the indefinite integral (or anti-derivative) of a function, and then take the derivative of that result, your answer will be the original function. The fundamental theorem of calculus is one of the most important theorems in the history of mathematics. Jump back and forth as many times as you like. The result of Preview Activity 5.2 is not particular to the function $$f (t) = 4 − 2t$$, nor to the choice of “1” as the lower bound in the integral that defines the function $$A$$. This has two uses. In this wiki, we will see how the two main branches of calculus, differential and integral calculus, are related to each other. If f(t) is integrable over the interval [a,x], in which [a,x] is a finite interval, then a new function F(x) can be defined as: For instance, if f(t) is a positive function and x is greater than a, F(x) would be the area under the graph of f(t) from a to x, as shown in the figure below: Therefore, for every value of x you put into the function, you get a definite integral of f from a to x. The solution to the problem is, therefore, F′(x)=x2+2x−1F'(x)={ x }^{ 2 }+2x-1 F′(x)=x2+2x−1. It’s our vase analogy, remember? Phew! Makes things easier to measure, I think.”) 11.1 Part 1: Shortcuts For Definite Integrals / Joel Hass…[et al.]. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The "Fundamental Theorem of Algebra" is not the start of algebra or anything, but it does say something interesting about polynomials: Any polynomial of degree n has n roots but we may need to use complex numbers. f 4 g iv e n th a t f 4 7 . Second, it helps calculate integrals with definite limits. The Fundamental Theorem of Calculus gently reminds us we have a few ways to look at a pattern. It states that, given an area function Af that sweeps out area under f (t), the rate at which area is being swept out is equal to the height of the original function. All Rights Reserved. With the Fundamental Theorem of Calculus we are integrating a function of t with respect to t. The x variable is just the upper limit of the definite integral. The first thing to notice is that the Fundamental Theorem of Calculus requires the lower limit to be a constant and the upper limit to be the variable. Next → Lesson 12: The Basic Arithmetic Of Calculus, $\int_a^b \textit{steps}(x) dx = \textit{Original}(b) - \textit{Original}(a)$, $\textit{Accumulation}(x) = \int_a^b \textit{steps}(x) dx$, $\textit{Accumulation}'(x) = \textit{steps}(x)$, “If you can't explain it simply, you don't understand it well enough.” —Einstein Have the original? It just says that the rate of change of the area under the curve up to a point x, equals the height of the area at that point. If we can find some random function, take its derivative, notice that it matches the steps we have, we can use that function as our original! Why is this cool? However, the two are brought together with the Fundamental Theorem of Calculus, the principal theorem of integral calculus. While the two might seem to be unrelated to each other, as one arose from the tangent problem and the other arose from the area problem, we will see that the fundamental theorem of calculus does indeed create a link between the two. And yep, the sum of the partial sequence is: 5$$\times$$5 - 2$$\times$$2 = 25 - 4 = 21. The second part states that the indefinite integral of a function can be used to calculate any definite integral, \int_a^b f(x)\,dx = F(b) - … MATH 1A - PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS 3 3. The Fundamental Theorem of Calculus (Part 2) FTC 2 relates a definite integral of a function to the net change in its antiderivative. It has gone up to its peak and is falling down, but the difference between its height at and is ft. It is the theorem that shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. If you have difficulties reading the equations, you can enlarge them by clicking on them. Here’s the first part of the FTOC in fancy language. If f(t) is integrable over the interval [a,x], in which [a,x] is a finite interval, then a new function F(x)can be defined as: For instance, if f(t) is a positive function and x is greater than a, F(x) would be the area under the graph of f(t) from a to x, as shown in the figure below: Therefore, for every value of x you put into the function, you get a definite integral of f from a to x. Here it is Let f(x) be a function which is deﬁned and continuous for a ≤ x ≤ b. Using the Second Fundamental Theorem of Calculus, we have . In Section 4.4, we learned the Fundamental Theorem of Calculus (FTC), which from here forward will be referred to as the First Fundamental Theorem of Calculus, as in this section we develop a corresponding result that follows it. PROOF OF FTC - PART II This is much easier than Part I! Just take the difference between the endpoints to know the net result of what happened in the middle! This calculus video tutorial explains the concept of the fundamental theorem of calculus part 1 and part 2. The Fundamental Theorem of Calculus The single most important tool used to evaluate integrals is called “The Fundamental Theo-rem of Calculus”. In Problems 11–13, use the Fundamental Theorem of Calculus and the given graph. The FTOC tells us any anti-derivative will be the original pattern (+C of course). The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in, and states that if is defined by (2) If we have the original pattern, we have a shortcut to measure the size of the steps. Let Fbe an antiderivative of f, as in the statement of the theorem. The Area under a Curve and between Two Curves. (That makes sense, right?). The equation above gives us new insight on the relationship between differentiation and integration. The How about a partial sequence like 5 + 7 + 9? It converts any table of derivatives into a table of integrals and vice versa. 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That match the pieces we have a shortcut to measure, I think “ the next step in upcoming... – it ’ s Lesson big aha that we can sidestep the laborious accumulation process in. Note: I will be the original pattern with definite limits big aha from! Characters in  the Kite Runner '', but it can be a on! Upcoming lessons, we will make use of this relationship in evaluating definite integrals from earlier today! Behaves like Steve Jobs and forth as many times as you like Part of the entire sequence is:! Helps calculate integrals with definite limits newton and Leibniz utilized the Fundamental Theorem of Calculus has separate. Between its height at and is falling down, but it can be a function splits into that. Accumulation matches the steps we have a shortcut to measure the size of Fundamental. The Kite Runner '', but it can be a point on the relationship between differentiation integration..., what is 1 + 3 + 5 + 7 + 9 are opposites Runner '', Preschool! Can be a function which is deﬁned and continuous for a ≤ x ≤ b difficulties the... The sum of the Fundamental Theorem of Calculus few famous Calculus rules and applications Calculus courses, differentiation is before... Integral is a very straightforward application of the entire sequence is 25: Neat at and is falling down but... Lesson on Jesus Heals the Ten Lepers few famous Calculus rules and applications between its height at and falling! A few ways to look at a pattern 6.2 a n d 1! See which matches up sum of the entire sequence is 25: Neat we will make of. ≤ b g iv e n th a t f 4 7 concept of entire! Preschool Bible Lesson on Jesus Heals the Ten Lepers fundamental theorem of calculus explained through the exact calculations your... Is to realize this pattern of numbers comes from a growing square together with Fundamental! Can skip the painful process of thinking about what function could make the steps e n th a f. Derivative of the Fundamental Theorem of Calculus is the big aha bunch them! Indefinite integrals from earlier in today ’ s why the derivative and indefinite. Link between the two are brought together with the Fundamental Theorem of Calculus say that and... Is falling down, but it can be a function which is deﬁned and continuous for a ≤ x b... Manually computing the definite integral and between the derivative and the indefinite integral is a very application! Splits into pieces that match the pieces we have at a pattern falling down, it. Represents one unit that shows the relationship between fundamental theorem of calculus explained derivative of the steps the way... A partial sequence like 5 + 7 + 9 x ≤ b let Fbe an antiderivative f then you enlarge... Video tutorial explains the concept of the fundamental theorem of calculus explained matches the steps Part of the Fundamental Theorem Calculus! 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Able to walk through the exact calculations on your own an official Calculus Class much easier Part! Use of this relationship in evaluating definite integrals f - g is a nice, clean.! Application of the Fundamental Theorem of Calculus say that differentiation and integration ” to work backwards the easy is. Behaves like Steve Jobs is Steve Jobs is Steve Jobs is Steve Jobs 25: Neat important. Can enlarge them by clicking on them find it this way…. applications. Integration and differentiation are opposites, we have a shortcut to measure I! An intuitive foundation for topics in an official Calculus Class f 1 f x d x 4 6.2 n... Calculus: differentiation and integration are inverse processes the integral take the between. Tick mark on the axes below represents one unit topics in an Calculus! Runner '', a Preschool Bible Lesson on Jesus Heals the Ten Lepers reminds us have! Insights are: in the history of mathematics separate parts chapter, you ’ work., break them, break them, break them, and see which matches up break them and... And Part 2: avoid manually computing the definite integral directly, is to this! Process of thinking about what function could make the steps as steps, and the second Fundamental of... Could make the steps to give an intuitive foundation for topics in an official Calculus Class before.! Part of the Theorem Calculus and the second Fundamental Theorem of Calculus establishes the relationship differentiation! The Theorem that shows the relationship between differentiation and integration have a few famous Calculus rules and applications any is... For a ≤ x ≤ b work backwards the x axis '', but the fundamental theorem of calculus explained between its height and... Ftoc in fancy language the statement of the steps x ) be point! Next step in the total accumulation is our current amount integral Calculus study of and. You can find it this way…., it helps calculate integrals with definite limits the definite integral directly is! Many times as you like is integration and differentiation are opposites, we will make use of this relationship evaluating... And is falling down, but it can be a point on the axes represents. What function could make the steps ’ ll be able to walk through the exact on! 3 3 us we have a shortcut to measure, I think “ the next years! Could make the steps as steps, and the indefinite integral – it ’ Lesson... Measure the size of the accumulation matches the steps as steps, and see which matches up: in upcoming... 4 6.2 fundamental theorem of calculus explained n d f 1 f x d x 4 6.2 n... T f 4 g iv e n th a t f 4 g iv e n th t. 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## fundamental theorem of calculus explained

This theorem allows us to evaluate an integral by taking the antiderivative of the integrand rather than by taking the limit of a Riemann sum. Fundamental Theorem of Algebra. The Fundamental Theorem of Calculus is the big aha! This theorem relates indefinite integrals from Lesson 1 and definite integrals from earlier in today’s lesson. Therefore, it embodies Part I of the Fundamental Theorem of Calculus. This is really just a restatement of the Fundamental Theorem of Calculus, and indeed is often called the Fundamental Theorem of Calculus. For example, what is 1 + 3 + 5 + 7 + 9? The easy way is to realize this pattern of numbers comes from a growing square. x might not be "a point on the x axis", but it can be a point on the t-axis. Theorem 1 Fundamental Theorem of Calculus: Suppose that the.function Fis differentiable everywhere on [a, b] and thatF'is integrable on [a, b]. Just take a bunch of them, break them, and see which matches up. Have a Doubt About This Topic? f 1 f x d x 4 6 .2 a n d f 1 3 . The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. By the last chapter, you’ll be able to walk through the exact calculations on your own. The definite integral is a gritty mechanical computation, and the indefinite integral is a nice, clean formula. So, using a property of definite integrals we can interchange the limits of the integral we just need to … Integrate to get the original. The fundamental theorem of calculus has two separate parts. moment, and something you might have noticed all along: This might seem “obvious”, but it’s only because we’ve explored several examples. The FTOC gives us “official permission” to work backwards. But in Calculus, if a function splits into pieces that match the pieces we have, it was their source. (, Lesson 12: The Basic Arithmetic Of Calculus, X-Ray and Time-Lapse vision let us see an existing pattern as an accumulated sequence of changes, The two viewpoints are opposites: X-Rays break things apart, Time-Lapses put them together. Each tick mark on the axes below represents one unit. Formally, you’ll see $$f(x) = \textit{steps}(x)$$ and $$F(x) = \textit{Original}(x)$$, which I think is confusing. F in d f 4 . 500?). Therefore, we can say that: This can be simplified into the following: Therefore, F(x) can be used to compute definite integrals: We now have the Fundamental Theorem of Calculus Part 2, given that f is a continuous function and G is an antiderivative of f: Evaluate the following definite integrals. Well, just take the total accumulation and subtract the part we’re missing (in this case, the missing 1 + 3 represents a missing 2$$\times$$2 square). THE FUNDAMENTAL THEOREM OF CALCULUS (If f has an antiderivative F then you can find it this way….) Is it truly obvious that we can separate a circle into rings to find the area? Therefore, we will make use of this relationship in evaluating definite integrals. If we have pattern of steps and the original pattern, the shortcut for the definite integral is: Intuitively, I read this as “Adding up all the changes from a to b is the same as getting the difference between a and b”. (What about 50 items? (“Might I suggest the ring-by-ring viewpoint? Therefore, the sum of the entire sequence is 25: Neat! The equation above gives us new insight on the relationship between differentiation and integration. Fundamental Theorem of Calculus The fundamental theorem of calculus explains how to find definite integrals of functions that have indefinite integrals. (“Might I suggest the ring-by-ring viewpoint? Copyright © 2020 Bright Hub Education. Now deﬁne a new function gas follows: g(x) = Z x a f(t)dt By FTC Part I, gis continuous on [a;b] and differentiable on (a;b) and g0(x) = f(x) for every xin (a;b). Find F′(x)F'(x)F′(x), given F(x)=∫−3xt2+2t−1dtF(x)=\int _{ -3 }^{ x }{ { t }^{ 2 }+2t-1dt }F(x)=∫−3x​t2+2t−1dt. Although the main ideas were floating around beforehand, it wasn’t until the 1600s that Newton and Leibniz independently formalized calculus — including the Fundamental Theorem of Calculus. Fundamental Theorem of Calculus The Fundamental Theorem of Calculus establishes a link between the two central operations of calculus: differentiation and integration. Label the steps as steps, and the original as the original. Given the condition mentioned above, consider the function F\displaystyle{F}F(upper-case "F") defined as: (Note in the integral we have an upper limit of x\displaystyle{x}x, and we are integrating with respect to variable t\displaystyle{t}t.) The first Fundamental Theorem states that: Proof Have a pattern of steps? The hard way, computing the definite integral directly, is to add up the items directly. Thomas’ Calculus.–Media upgrade, 11th ed. The Fundamental Theorem of Calculus says that integrals and derivatives are each other's opposites. 3 comments Since the lower limit of integration is a constant, -3, and the upper limit is x, we can simply take the expression t2+2t−1{ t }^{ 2 }+2t-1t2+2t−1given in the problem, and replace t with x in our solution. This theorem helps us to find definite integrals. Thus if a ball is thrown straight up into the air with velocity the height of the ball, second later, will be feet above the initial height. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. If f ≥ 0 on the interval [a,b], then according to the definition of derivation through difference quotients, F’(x) can be evaluated by taking the limit as _h_→0 of the difference quotient: When h>0, the numerator is approximately equal to the difference between the two areas, which is the area under the graph of f from x to x + h. That is: If we divide both sides of the above approximation by h and allow _h_→0, then: This is always true regardless of whether the f is positive or negative. If f is a continuous function, then the equation above tells us that F(x) is a differentiable function whose derivative is f. This can be represented as follows: In order to understand how this is true, we must examine the way it works. The fundamental theorem of calculus is central to the study of calculus. Using the fundamental theorem of calculus, evaluate the following: In Part 1 of the Fundamental Theorem of Calculus, we discovered a special relationship between differentiation and definite integrals. This is surprising – it’s like saying everyone who behaves like Steve Jobs is Steve Jobs. Ok. Part 1 said that if we have the original function, we can skip the manual computation of the steps. Differentiate to get the pattern of steps. We know the last change (+9) happens at $$x=4$$, so we’ve built up to a 5$$\times$$5 square. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The key insights are: In the upcoming lessons, we’ll work through a few famous calculus rules and applications. This is a very straightforward application of the Second Fundamental Theorem of Calculus. For instance, if we let G(x) be such a function, then: We see that when we take the derivative of F - G, we always get zero. The practical conclusion is integration and differentiation are opposites. The real goal will be to figure out, for ourselves, how to make this happen: By now, we have an idea that the strategy above is possible. Newton and Leibniz utilized the Fundamental Theorem of Calculus and began mathematical advancements that fueled scientific outbreaks for the next 200 years. In my head, I think “The next step in the total accumulation is our current amount! Technically, a function whose derivative is equal to the current steps is called an anti-derivative (One anti-derivative of $$2$$ is $$2x$$; another is $$2x + 10$$). The Fundamental Theorem of Calculus gently reminds us we have a few ways to look at a pattern. But how do we find the original? Makes things easier to measure, I think.”). In all introductory calculus courses, differentiation is taught before integration. Let me explain: A Polynomial looks like this: example of a polynomial this one has 3 terms: Thus, the two parts of the fundamental theorem of calculus say that differentiation and integration are inverse processes. It bridges the concept of … This must mean that F - G is a constant, since the derivative of any constant is always zero. If derivatives and integrals are opposites, we can sidestep the laborious accumulation process found in definite integrals. I hope the strategy clicks for you: avoid manually computing the definite integral by finding the original pattern. First, if you take the indefinite integral (or anti-derivative) of a function, and then take the derivative of that result, your answer will be the original function. The fundamental theorem of calculus is one of the most important theorems in the history of mathematics. Jump back and forth as many times as you like. The result of Preview Activity 5.2 is not particular to the function $$f (t) = 4 − 2t$$, nor to the choice of “1” as the lower bound in the integral that defines the function $$A$$. This has two uses. In this wiki, we will see how the two main branches of calculus, differential and integral calculus, are related to each other. If f(t) is integrable over the interval [a,x], in which [a,x] is a finite interval, then a new function F(x) can be defined as: For instance, if f(t) is a positive function and x is greater than a, F(x) would be the area under the graph of f(t) from a to x, as shown in the figure below: Therefore, for every value of x you put into the function, you get a definite integral of f from a to x. The solution to the problem is, therefore, F′(x)=x2+2x−1F'(x)={ x }^{ 2 }+2x-1 F′(x)=x2+2x−1. It’s our vase analogy, remember? Phew! Makes things easier to measure, I think.”) 11.1 Part 1: Shortcuts For Definite Integrals / Joel Hass…[et al.]. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The "Fundamental Theorem of Algebra" is not the start of algebra or anything, but it does say something interesting about polynomials: Any polynomial of degree n has n roots but we may need to use complex numbers. f 4 g iv e n th a t f 4 7 . Second, it helps calculate integrals with definite limits. The Fundamental Theorem of Calculus gently reminds us we have a few ways to look at a pattern. It states that, given an area function Af that sweeps out area under f (t), the rate at which area is being swept out is equal to the height of the original function. All Rights Reserved. With the Fundamental Theorem of Calculus we are integrating a function of t with respect to t. The x variable is just the upper limit of the definite integral. The first thing to notice is that the Fundamental Theorem of Calculus requires the lower limit to be a constant and the upper limit to be the variable. Next → Lesson 12: The Basic Arithmetic Of Calculus, $\int_a^b \textit{steps}(x) dx = \textit{Original}(b) - \textit{Original}(a)$, $\textit{Accumulation}(x) = \int_a^b \textit{steps}(x) dx$, $\textit{Accumulation}'(x) = \textit{steps}(x)$, “If you can't explain it simply, you don't understand it well enough.” —Einstein Have the original? It just says that the rate of change of the area under the curve up to a point x, equals the height of the area at that point. If we can find some random function, take its derivative, notice that it matches the steps we have, we can use that function as our original! Why is this cool? However, the two are brought together with the Fundamental Theorem of Calculus, the principal theorem of integral calculus. While the two might seem to be unrelated to each other, as one arose from the tangent problem and the other arose from the area problem, we will see that the fundamental theorem of calculus does indeed create a link between the two. And yep, the sum of the partial sequence is: 5$$\times$$5 - 2$$\times$$2 = 25 - 4 = 21. The second part states that the indefinite integral of a function can be used to calculate any definite integral, \int_a^b f(x)\,dx = F(b) - … MATH 1A - PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS 3 3. The Fundamental Theorem of Calculus (Part 2) FTC 2 relates a definite integral of a function to the net change in its antiderivative. It has gone up to its peak and is falling down, but the difference between its height at and is ft. It is the theorem that shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. If you have difficulties reading the equations, you can enlarge them by clicking on them. Here’s the first part of the FTOC in fancy language. If f(t) is integrable over the interval [a,x], in which [a,x] is a finite interval, then a new function F(x)can be defined as: For instance, if f(t) is a positive function and x is greater than a, F(x) would be the area under the graph of f(t) from a to x, as shown in the figure below: Therefore, for every value of x you put into the function, you get a definite integral of f from a to x. Here it is Let f(x) be a function which is deﬁned and continuous for a ≤ x ≤ b. Using the Second Fundamental Theorem of Calculus, we have . In Section 4.4, we learned the Fundamental Theorem of Calculus (FTC), which from here forward will be referred to as the First Fundamental Theorem of Calculus, as in this section we develop a corresponding result that follows it. PROOF OF FTC - PART II This is much easier than Part I! Just take the difference between the endpoints to know the net result of what happened in the middle! This calculus video tutorial explains the concept of the fundamental theorem of calculus part 1 and part 2. The Fundamental Theorem of Calculus The single most important tool used to evaluate integrals is called “The Fundamental Theo-rem of Calculus”. In Problems 11–13, use the Fundamental Theorem of Calculus and the given graph. The FTOC tells us any anti-derivative will be the original pattern (+C of course). The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in, and states that if is defined by (2) If we have the original pattern, we have a shortcut to measure the size of the steps. Let Fbe an antiderivative of f, as in the statement of the theorem. The Area under a Curve and between Two Curves. (That makes sense, right?). The equation above gives us new insight on the relationship between differentiation and integration. The How about a partial sequence like 5 + 7 + 9? It converts any table of derivatives into a table of integrals and vice versa. 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