5. Integration

Chapter 5 Review Exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous over their domains.

1. If f(x)>0,{f}^{\prime }(x)>0 for all x, then the right-hand rule underestimates the integral {\int }_{a}^{b}f(x). Use a graph to justify your answer.

Solution

False

2. {\int }_{a}^{b}f{(x)}^{2}dx={\int }_{a}^{b}f(x)dx{\int }_{a}^{b}f(x)dx

3. If f(x)\le g(x) for all x\in \left[a,b\right], then {\int }_{a}^{b}f(x)\le {\int }_{a}^{b}g(x).

Solution

True

4. All continuous functions have an antiderivative.

Evaluate the Riemann sums {L}_{4}\text{ and }{R}_{4} for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5. y=3{x}^{2}-2x+1 over \left[-1,1\right]

Solution

{L}_{4}=5.25,{R}_{4}=3.25, exact answer: 4

6. y=\text{ln}({x}^{2}+1) over \left[0,e\right]

7. y={x}^{2} \sin x over \left[0,\pi \right]

Solution

{L}_{4}=5.364,{R}_{4}=5.364, exact answer: 5.870

8. y=\sqrt{x}+\frac{1}{x} over \left[1,4\right]

Evaluate the following integrals.

9. {\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx

Solution

-\frac{4}{3}

10. {\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt

11. {\int }_{\pi \text{/}3}^{\pi \text{/}2}2 \sec (2\theta ) \tan (2\theta )d\theta

Solution

1

12. {\int }_{0}^{\pi \text{/}4}{e}^{{ \cos }^{2}x} \sin x \cos dx

Find the antiderivative.

13. \int \frac{dx}{{(x+4)}^{3}}

Solution

-\frac{1}{2{(x+4)}^{2}}+C

14. \int x\text{ln}({x}^{2})dx

15. \int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx

Solution

\frac{4}{3}\phantom{\rule{0.05em}{0ex}}{ \sin }^{-1}({x}^{3})+C

16. \int \frac{{e}^{2x}}{1+{e}^{4x}}dx

Find the derivative.

17. \frac{d}{dt}{\int }_{0}^{t}\frac{ \sin x}{\sqrt{1+{x}^{2}}}dx

Solution

\frac{ \sin t}{\sqrt{1+{t}^{2}}}

18. \frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt

19. \frac{d}{dx}{\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt

Solution

4\frac{\text{ln}x}{x}+1

20. \frac{d}{dx}{\int }_{0}^{ \cos x}{e}^{{t}^{2}}dt

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($)
1980 0
1985 −5,468,750
1990 755,495
1995 −73,005
2000 −29,768
2005 −918
2010 −177

21. If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

Solution

$6,328,113

22. The average cost per gigabyte of RAM can be approximated by the function C(t)=8,500,000{(0.65)}^{t}, where t is measured in years since 1980, and C is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23. Find the average cost of 1GB RAM for 2005 to 2010.

Solution

$73.36

24. The velocity of a bullet from a rifle can be approximated by v(t)=6400{t}^{2}-6505t+2686, where t is seconds after the shot and v is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: 0\le t\le 0.5. What is the total distance the bullet travels in 0.5 sec?

25. What is the average velocity of the bullet for the first half-second?

Solution

\frac{19117}{12}\text{ft/sec},\text{or}1593\text{ft/sec}

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