Practice Problems
Work through the odd-numbered problems 1-41. Once you have completed the problem set, check your answers.
Practice Problems
In problems 1 – 3, use the values in Table 1 to estimate the areas.
| x | f(x) | g(x) | h(x) |
|---|---|---|---|
| 0 | 5 | 2 | 5 |
| 1 | 6 | 1 | 6 |
| 2 | 6 | 2 | 8 |
| 3 | 4 | 2 | 6 |
| 4 | 3 | 3 | 5 |
| 5 | 2 | 4 | 4 |
| 6 | 2 | 5 | 2 |
1. Estimate the area between \(f\) and \(g\) for \(1 \leq \mathrm{x} \leq 4\).
3. Estimate the area between \(f\) and \(h\) for \(0 \leq x \leq 4\).
5. Estimate the area of the island in Fig. 13.
In problems 7 – 17, sketch the graph of each function and find the area between the graphs of \(f\) and \(g\) for \(x\) in the given interval.
7. \(\mathrm{f}(x)=x^{2}+3, \mathrm{~g}(x)=1\) and \(-1 \leq x \leq 2\).
9. \(\mathrm{f}(x)=x^{2}, \mathrm{~g}(x)=x \) and \(0 \leq x \leq 2\).
11. \(\mathrm{f}(x)=\frac{1}{x}, \mathrm{~g}(x)=x\) and \(1 \leq x \leq \mathrm{e}\).
13. \(\mathrm{f}(x)=x+1, \mathrm{~g}(x)=\cos (x)\) and \(0 \leq x \leq \pi / 4\).
15. \(f(x)=\mathrm{e}^{x}, \mathrm{~g}(x)=x\) and \(0 \leq x \leq 2\).
17. \(f(x)=3, g(x)=\sqrt{1-x^{2}}\) and \(0 \leq x \leq 1\).
In problems 19 – 21, use the values in Table 1 to estimate the average values.
19. Estimate the average value of \(f\) on the interval \([0.5, 4.5]\).
21. Estimate the average value of \(f\) on the interval \([1.5, 3.5]\).
In problems 23 – 31, find the average value of \(f\) on the given interval.
23. \(f(x)\) in Fig. 14 for \(0 \leq x \leq 2\).
25. \(f(x)\) in Fig. 14 for\(1 \leq x \leq 6\).
27. \(\mathrm{f}(x)=2 x+1\) for \(0 \leq x \leq 4\).
29. \(\mathrm{f}(x)=x^{2} \) for \(1 \leq x \leq 3\).
31. \(\begin{aligned} &\mathrm{f}(x)=\sin (x) \text { for } 0 \leq x \leq \pi \\ \end{aligned}\).
33. Calculate the average value of \(\mathrm{f}(x)=\sqrt{x}\) on the interval \([0, C]\) for \(C = 1, 9, 81, 100\). What is the pattern?
35. Fig. 15 shows the number of telephone calls per minute at a large company. (a) Estimate the average number of calls per minute from 8 am to 5 pm. (b) From 9 am to 1 pm.
37. (a) How much work is done lifting a 20 pound bucket from the ground to the top of a 30 foot building with a cable which weighs 3 pounds per foot? (b) How much work is done lifting the same bucket from the ground to a height of 15 feet with the same cable?
39. (a) How much work is done lifting a 10 pound calculus book from the ground to the top of a 30 foot building with a cable which weighs 2 pounds per foot? (b) From the ground to a height of 10 feet? (c) From a height of 10 feet to a height of 20 feet?
41. How much work is done lifting an 60 pound injured child to the top of a 15 foot hole using a stretcher weighing 10 pounds and a cable which weighs 2 pound per foot?
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.8-First-Applications-of-Definite-Integrals.pdf
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