1. 5
INTEGRALS

2. 5.3 The Fundamental Theorem of Calculus
INTEGRALS
5.3 The Fundamental Theorem of Calculus
In this section, we will learn about:
The Fundamental Theorem of Calculus
and its significance.

3. FUNDAMENTAL THEOREM OF CALCULUS
The Fundamental Theorem of Calculus (FTC) is appropriately named.
It establishes a connection between the two branches of calculus—differential calculus and integral calculus.

4. Differential calculus arose from the tangent problem.
FTC
Differential calculus arose from the tangent problem.
Integral calculus arose from a seemingly unrelated problem—the area problem.

5. FTC
The FTC gives the precise inverse relationship between the derivative and the integral.

6. FTC
Equation 1
The first part of the FTC deals with functions defined by an equation of the form
where f is a continuous function on [a, b] and x varies between a and b.

7. FTC
Observe that g depends only on x, which appears as the variable upper limit in the integral.
If x is a fixed number, then the integral is a definite number.
If we then let x vary, the number also varies and defines a function of x denoted by g(x).

8. FTC
If f happens to be a positive function, then g(x) can be interpreted as the area under the graph of f from a to x, where x can vary from a to b.
Think of g as the ‘area so far’ function, as seen here.

9. FTC
Example 1
If f is the function whose graph is shown and , find the values of: g(0), g(1), g(2), g(3), g(4), and g(5).
Then, sketch a rough graph of g.

10. FTC
Example 1
First, we notice that:

11. From the figure, we see that g(1) is the area of a triangle:
FTC
Example 1
From the figure, we see that g(1) is the area of a triangle:

12. To find g(2), we add to g(1) the area of a rectangle:
FTC
Example 1
To find g(2), we add to g(1) the area of a rectangle:

13. We estimate that the area under f from 2 to 3 is about 1.3.
FTC
Example 1
We estimate that the area under f from 2 to 3 is about 1.3.
So,

14. For t > 3, f(t) is negative.
FTC
Example 1
For t > 3, f(t) is negative.
So, we start subtracting areas, as follows.

15. FTC
Example 1
Thus,

16. We use these values to sketch the graph of g.
FTC
Example 1
We use these values to sketch the graph of g.
Notice that, because f(t) is positive for t < 3, we keep adding area for t < 3.
So, g is increasing up to x = 3, where it attains a maximum value.
For x > 3, g decreases because f(t) is negative.

17. FTC
If we take f(t) = t and a = 0, then, using Exercise 27 in Section 5.2, we have:

18. FTC
If we sketch the derivative of the function g, as in the first figure, by estimating slopes of tangents, we get a graph like that of f in the second figure.
So, we suspect that g’ = f in Example 1 too.

19. FTC
To see why this might be generally true, we consider a continuous function f with f(x) ≥ 0.
Then, can be interpreted as the area under the graph of f from a to x.

20. If f is continuous on [a, b], then the function g defined by
FTC1
If f is continuous on [a, b], then the function g defined by
is continuous on [a, b] and differentiable on (a, b), and g’(x) = f(x).

21. FTC1
In words, the FTC1 says that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit.

22. Using Leibniz notation for derivatives, we can write the FTC1 as
Equation 5
Using Leibniz notation for derivatives, we can write the FTC1 as
when f is continuous.
Roughly speaking, Equation 5 says that, if we first integrate f and then differentiate the result, we get back to the original function f.

23. Find the derivative of the function
FTC1
Example 2
Find the derivative of the function
As is continuous, the FTC1 gives:

24. FTC1
Example 3
A formula of the form may seem like a strange way of defining a function.
However, books on physics, chemistry, and statistics are full of such functions.

25. FTC1
Example 4
Find
Here, we have to be careful to use the Chain Rule in conjunction with the FTC1.

26. FTC1
Example 4
Let u = x4.
Then,

27. FTC1
In Section 5.2, we computed integrals from the definition as a limit of Riemann sums and saw that this procedure is sometimes long and difficult.
The second part of the FTC (FTC2), which follows easily from the first part, provides us with a much simpler method for the evaluation of integrals.

28. If f is continuous on [a, b], then
FTC2
If f is continuous on [a, b], then
where F is any antiderivative of f, that is, a function such that F’ = f.

29. FTC2
The FTC2 states that, if we know an antiderivative F of f, then we can evaluate simply by subtracting the values of F at the endpoints of the interval [a, b].

30. FTC2
It’s very surprising that , which was defined by a complicated procedure involving all the values of f(x) for a ≤ x ≤ b, can be found by knowing the values of F(x) at only two points, a and b.

31. So, s is an antiderivative of v.
FTC2
If v(t) is the velocity of an object and s(t) is its position at time t, then v(t) = s’(t).
So, s is an antiderivative of v.

32. FTC2
In Section 5.1, we considered an object that always moves in the positive direction.
Then, we guessed that the area under the velocity curve equals the distance traveled.
In symbols,
That is exactly what the FTC2 says in this context.

33. FTC2
Example 5
Evaluate the integral
The function f(x) = x3 is continuous on [-2, 1] and we know from Section 4.9 that an antiderivative is F(x) = ¼x4.
So, the FTC2 gives:

34. Notice that the FTC2 says that we can use any antiderivative F of f.
Example 5
Notice that the FTC2 says that we can use any antiderivative F of f.
So, we may as well use the simplest one, namely F(x) = ¼x4, instead of ¼x or ¼x4 + C.

35. We often use the notation
FTC2
We often use the notation
So, the equation of the FTC2 can be written as:
Other common notations are and

36. Find the area under the parabola y = x2 from 0 to 1.
FTC2
Example 6
Find the area under the parabola y = x2 from 0 to 1.
An antiderivative of f(x) = x2 is F(x) = (1/3)x3.
The required area is found using the FTC2:

37. Find the area under the cosine curve from 0 to b, where 0 ≤ b ≤ π/2.
FTC2
Example 7
Find the area under the cosine curve from 0 to b, where 0 ≤ b ≤ π/2.
Since an antiderivative of f(x) = cos x is F(x) = sin x, we have:

38. FTC2
Example 7
In particular, taking b = π/2, we have proved that the area under the cosine curve from 0 to π/2 is sin(π/2) =1.

39. FTC2
If we didn’t have the benefit of the FTC, we would have to compute a difficult limit of sums using either:
Obscure trigonometric identities
A computer algebra system (CAS), as in Section 5.1

40. What is wrong with this calculation?
FTC2
Example 8
What is wrong with this calculation?

41. FTC2
Example 9
To start, we notice that the calculation must be wrong because the answer is negative but f(x) = 1/x2 ≥ 0 and Property 6 of integrals says that when f ≥ 0.

42. The FTC applies to continuous functions.
Example 9
The FTC applies to continuous functions.
It can’t be applied here because f(x) = 1/x2 is not continuous on [-1, 3].
In fact, f has an infinite discontinuity at x = 0.
So, does not exist.

43. We end this section by bringing together the two parts of the FTC.
INVERSE PROCESSES
We end this section by bringing together the two parts of the FTC.

44. Suppose f is continuous on [a, b].
FTC
Suppose f is continuous on [a, b].
1.If , then g’(x) = f(x).
, where F is any antiderivative of f, that is, F’ = f.

45. We noted that the FTC1 can be rewritten as:
INVERSE PROCESSES
We noted that the FTC1 can be rewritten as:
This says that, if f is integrated and then the result is differentiated, we arrive back at the original function f.

46. As F’(x) = f(x), the FTC2 can be rewritten as:
INVERSE PROCESSES
As F’(x) = f(x), the FTC2 can be rewritten as:
This version says that, if we take a function F, first differentiate it, and then integrate the result, we arrive back at the original function F.
However, it’s in the form F(b) - F(a).

47. INVERSE PROCESSES
Taken together, the two parts of the FTC say that differentiation and integration are inverse processes.
Each undoes what the other does.

48. The FTC is unquestionably the most important theorem in calculus.
SUMMARY
The FTC is unquestionably the most important theorem in calculus.
Indeed, it ranks as one of the great accomplishments of the human mind.