Chapter 4: The Integral
Chapter 4 Review Exercises
From Calculus, Volume 1, by Strang and Herman, OpenStax (Web), licensed under a CC BY-NC-SA 4.0 License.
True or false? (Assume all functions [latex]f[/latex] and [latex]g[/latex] are continuous over their domains.)
1. If [latex]f(x) < 0,{f}^{\prime }(x) < 0[/latex] for all [latex]x[/latex], then the right-hand rule underestimates the integral [latex]{\int }_{a}^{b}f(x)[/latex]. Use a graph to justify your answer.
Solution
False
2. [latex]{\int }_{a}^{b}f{(x)}^{2}dx={\int }_{a}^{b}f(x)dx{\int }_{a}^{b}f(x)dx[/latex]
3. If [latex]f(x)\le g(x)[/latex] for all [latex]x\in \left[a,b\right][/latex], then [latex]{\int }_{a}^{b}f(x)\le {\int }_{a}^{b}g(x)[/latex].
Solution
True
4. All continuous functions have an antiderivative.
Evaluate the Riemann sums [latex]{L}_{4}\text{ and }{R}_{4}[/latex] 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. [latex]y=3{x}^{2}-2x+1[/latex] over [latex]\left[-1,1\right][/latex]
Solution
[latex]{L}_{4}=5.25,{R}_{4}=3.25[/latex], exact answer: 4
6. [latex]y=\sqrt{x}+\frac{1}{x}[/latex] over [latex]\left[1,4\right][/latex]
Evaluate the following integrals.
7. [latex]{\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx[/latex]
Solution
[latex]-\frac{4}{3}[/latex]
8. [latex]{\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt[/latex]
Find the antiderivative using substitution.
9. [latex]\int \frac{dx}{{(x+4)}^{3}}[/latex]
Solution
[latex]-\frac{1}{2{(x+4)}^{2}}+C[/latex]
Find the derivative.
10. [latex]\frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt[/latex]
11. [latex]\frac{d}{dx}{\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt[/latex]
Solution
[latex]4\frac{\text{ln}x}{x}+1[/latex]
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 |
12. If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.
Solution
$6,328,113
13. The average cost per gigabyte of RAM can be approximated by the function [latex]C(t)=8,500,000{(0.65)}^{t}[/latex], where [latex]t[/latex] is measured in years since 1980, and [latex]C[/latex] is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.
14. Find the average cost of 1GB RAM for 2005 to 2010.
Solution
$73.36
15. The velocity of a bullet from a rifle can be approximated by [latex]v(t)=6400{t}^{2}-6505t+2686[/latex], where [latex]t[/latex] is seconds after the shot and [latex]v[/latex] is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: [latex]0\le t\le 0.5[/latex]. What is the total distance the bullet travels in 0.5 sec?
16. What is the average velocity of the bullet for the first half-second?
Solution
[latex]\frac{19117}{12}\text{ft/sec},\text{or}1593\text{ft/sec}[/latex]