The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

SUGGESTED REFERENCE MATERIAL:

As you work through the problems listed below, you should reference Chapter 5.6 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.

EXPECTED SKILLS:

? Be able to use one part of the Fundamental Theorem of Calculus (FTC) to evaluate definite integrals via antiderivatives.

? Know how to use another part of the FTC to compute derivatives of functions defined as integrals.

PRACTICE PROBLEMS:

1 1. Consider the graph of f (x) = x - 1 on [1, 4], shown below.

2

(a) Use a definite intergal and the Fundamental Theorem of Calculus to compute the net signed area between the graph of f (x) and the x-axis on the interval [1, 4].

41

3

x - 1 dx =

12

4

(b) Verify your answer from part (a) by using appropriate formulae from geometry.

1

Alower

triangle

=

; 4

Aupper

triangle

=

1;

3

Thus, the value of the definite integral is -Alower triangle + Aupper triangle = 4

1

For problems 2-4, sketch a region whose net signed area is equivalent to the value of the given definite integral. Then evaluate the definite integral using any method.

8

2. (x2 - 4x - 5) dx

0

3 2

3. cos x dx

2

8

(x2

- 4x - 5) dx

=

8

0

3

-1 2 4. -4 x3 dx

3 2

cos x dx = -2

2

-1 2

15

-4

x3

dx

=

- 16

2

For problems 5-15, evaluate the given definite integral.

25

1 5. dx

xx

4

3

5

-1 x + 1

6.

dx

-e x

-2 + e; Detailed Solution: Here

ln 3

7.

e2x dx

ln 2

5

2

2 3

8. csc (x) cot (x) dx

2

1 - 2 3

3

3

9.

dx

1 + x2

0

9

10. |x - 5| dx

-9

106

e6

1

11.

dx

10x

1

3

5

3

3

2

1

12.

dx

1

1 - x2

2

12

13. | cos x| dx

0

2; Video Solution:

3

14. f (x) dx if f (x) =

0

51 2

x + 5 if x 1 4x + 2 if x > 1

15. 4 tan2 x dx. (HINT: Use a trigonometric identity first to rewrite the integrand.)

0

1-

4

16. Definitions: If an object moves along a straight line with position function s(t), its velocity function is v(t) = s (t). Then:

? The displacement from time t1 to time t2 is the net change of position of the parti-

t2

cle during the time period from t1 to t2 and is calculated by evaluating v(t) dt.

t1

? The total distance traveled from time t1 to time t2 is calculated by evaluating t2 |v(t)| dt. t1

Assume that a particle is moving along a straight line such that its velocity at time t is v(t) = t2 - 6t + 5 (meters per second).

(a) Compute the displacement of the particle during the time period 0 t 6. -6 meters

(b) Compute the total distance traveled by the particle during the time period 0 t 6. 46 meters 3

4

17. The following Riemann Sum was derived by dividing an interval [a, b] into n subintervals of equal width and then choosing xk to be the right endpoint of each subinterval.

n

44

lim

1+ k

n+

nn

k=1

(a) What is the interval, [a, b]? If we consider f (x) = x, then the interval is [1, 5]

(b) Convert the Riemann Sum to an equivalent definite integral.

n

44

5

lim

1 + k = x dx

n+ k=1

nn 1

(c) Using the definite integral from part (b) and part of the Fundamental Theorem of Calculus, evaluate the limit.

12

NOTE: In number 17, we could have considered f (x) = 1 + x. In that case, [a, b] =

n

44

4

[0, 4] and lim

1 + k = (1 + x) dx. The value of this definite integral is

n+ k=1

nn 0

also 12.

18. Explain what is wrong with the following calculation:

11

1 x=1

-1

x2

dx

=

- x

x=-1

=

-1 - (1)

=

-2

1 f (x) = is not continuous at x = 0 which is in [-1, 1]; so, the FTC does not

x2 immediately apply.

For problems 19-22, use part of the Fundamental Theorem of Calculus to compute the indicated derivative.

x

d

19.

ln (t) dt

dx

2

ln (x)

10

d 20.

et2 dt

dx

x

-ex2

5

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