There Is a Relationship Between Integrals and Derivatives That Most See As Shocking

Problem Set 17 – Fundamental Theorem of Calculus

Problem Set 17 – The Fundamental Theorem of Calculus

17.1 The Fundamental Theorem of Calculus provides a way of computing definite integrals using antiderivatives. Let’s see how this works.

1) Let’s revisit problems 3 and 4 from Problem Set 16. 2 a) The area function for problem 3 is A0 (x)  x  4x . Find the derivative of this area function. What is the relationship between this derivative and f (x)  2x  4 , the line graphed in problem 3?

We say that “ A 0 (x) is an ______-derivative of f (x) .”

2 b) The area function for problem 4 is A1 (x)  x  4x  5 . Find the derivative of this area function. What is the relationship between this derivative and f (x)  2x  4 , the line graphed in problem 4?

We can also say that “ A1 (x) is an ______of f (x) .”

c) What is the difference between A 0 and A1 ? How is this represented in the graphs we shaded in parts a) of problems 3 and 4 of Problem Set 16?

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1 Problem Set 17 – Fundamental Theorem of Calculus

6 A (x)  x 2  4x (2x  4)dx d) Use the area function 0 to find the area represented by  . (Hint: 4 consider subtracting areas.) Draw a picture that shows how this works.

6 A (x) x 2 4x 5 (2x  4)dx e) Use the area function 1    to find the area represented by  . 4 Draw a picture that shows how this works.

f) We saw in parts a and b above that A 0 (x) and A1 (x) are both anti-derivatives of

f (x)  2x  4 . Find another anti-derivative for f and call it A 2 (x) .

g) Compute A 2 (6)  A2 (4) . What do you notice about this difference? (Hint: look at parts d and e.)

6 h) Make a conjecture about  (2x  4)dx . 4 6 If F is an antiderivative of f, then  (2x  4)dx = ______. 4

2) We would like to compute the area under the graph of y f (x)  x3  8x 2 16x  3 on the interval [1,5]. Its graph is provided.

a) Write an expression for this area as a definite integral.

1 2 3 4 5 x

b) Find an anti-derivative of f (x) , and name this function F(x) .

c) Compute F(1) and F(5) .

d) By considering how your conjecture could also apply to this case, compute the area shown on the graph. Is your answer reasonable?

3) Let’s make our conjecture more general. If F(x) is an anti-derivative for f (x) , then b  f (x)dx = ______. This is called the Fundamental a Theorem of Calculus.

This means that to evaluate a definite integral, you find an antiderivative, evaluate it at the endpoints and calculate the difference. A common notation for this is b f (x) dx  F(x) b  F(b)  F(a).  a a

6 Here’s an example of this notation. We’ll compute  (2x  4)dx as follows: 4 6 6 (2x  4)dx  x 2  4x  62  4  6 42  4  4.  4 4

4) Let’s practice using the Fundamental Theorem of Calculus. Compute

6 a) ( t 2  3 t ) d t 0

3 b) (2t 3  7)dt 1

5 c) 4  et dt 0

y d) Compute the shaded area using the Fundamental Theorem of Calculus.

1 1 2 3 4 5 x

5) Let’s consider functions that take on negative values. 4  1  a) Using the Fundamental Theorem of Calculus, compute  x  2 dx 1 2 

b) Since your answer to a) is negative, it can’t represent an area. However, there is a connection to area. What is it? It may help to graph the function.

7  1  c) Now compute  x  2 dx 1 2 

d) Since your answer to c) is negative, it also can’t represent an area. However, there is still a connection to area. What is it? (Do you see a connection between this problem and #1g on Problem Set 11?)

6) f A

x1 x2 B x3 x

a) Express the total shaded area using definite integrals.

x 3 b) Express  f ( x ) d x in terms of the area of A and the area of B. x 1

17.2 In light of the fact that areas can be represented by definite integrals, and definite integrals can be computed from antiderivatives using the Fundamental Theorem of Calculus, we’ll introduce a new notation for antiderivatives.

As we have seen earlier, a function such as f (x)  5x 4 has many antiderivatives, including x5 11, x5  2 , x5  3871, etc., so we write

5x4dx  x5  C , where C can be any real number.

We will also refer to antiderivatives as indefinite integrals.

7) Compute the following indefinite integrals: a)  2x dx

b) (2x  4)dx

c) 60dt

d)  (w2  3)dw

e)  (x 99  8x 7  35x 4 )dx

f)  5er dr

8) Alisa was traveling 60 mph for 5 hours when she suddenly hit a yellow frog.

a) Set up the definite integral for the distance traveled during the 5 hours.

b) Using the Fundamental Theorem of Calculus, compute the distance traveled.

9) After giving the frog a proper burial, Alisa returned to the freeway with a velocity of v(t)  10t (in feet per second). How far had she gone in the first 4 seconds?

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 7