3.7.E: Exercises
- Page ID
- 157591
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In problems 1 – 4, use the values in the table 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 |
0 |
2 |
1. Estimate the area between \(f\) and \(g\), between \(x = 0\) and \(x = 4\).
2. Estimate the area between \(g\) and \(h\), between \(x = 0\) and \(x = 6\).
3. Estimate the area between \(f\) and \(h\), between \(x = 0\) and \(x = 4\).
4. Estimate the area between \(f\) and \(g\), between \(x = 0\) and \(x = 6\).
Estimate the area of the island shown
In problems 6 – 15, find the area between the graphs of \(f\) and \(g\) for \(x\) in the given interval. Remember to draw the graph!
| 6. \(f(x) = x^2 + 3 \), \(g(x) = 1\) and \(–1 \leq x \leq 2\). |
| 7. \(f(x) = x^2 + 3 \), \(g(x) = 1 + x\) and \(0 \leq x \leq 3\). |
| 8. \(f(x) = x^2 \), \(g(x) = x\) and \(0 \leq x \leq 2\). |
| 9. \(f(x) = (x –1)^2 \), \(g(x) = x + 1\) and \(0 \leq x \leq 3\). |
| 10. \(f(x) = \frac{1}{x}\), \(g(x) = x\) and \(1 \leq x \leq e\). |
| 11. \(f(x) = \sqrt{x}\), \(g(x) = x\) and \(0 \leq x \leq 4\). |
| 12. \(f(x) = 4 – x^2 \), \(g(x) = x + 2\) and \(0 \leq x \leq 2\). |
| 13. \(f(x) = e^x\), \(g(x) = x\) and \(0 \leq x \leq 2\). |
| 14. \(f(x) = 3 \), \(g(x) = \sqrt{1-x^2}\) and \(0 \leq x \leq 1\). |
| 15. \(f(x) = 2 \), \(g(x) = \sqrt{4-x^2}\) and \(–2 \leq x \leq 2\). |
For problems 16-18, find the volume of the solid obtained by rotating the specified region about the \(x\) axis.
16. Region under \(f(x) = x^2 + 3\) for \(–1 \leq x \leq 2\).
17. Region under \(f(x) = 4 – x^2\) for \(0 \leq x \leq 2\).
18. Region under \(f(x) = \frac{1}{x}\) for \(1 \leq x \leq 2\).
In problems 19 and 20 use the values in the table to estimate the average values.
| \(x\) | \(f(x)\) | \(g(x)\) |
|
0 |
5 |
2 |
|
1 |
6 |
1 |
|
2 |
6 |
2 |
|
3 |
4 |
2 |
|
4 |
3 |
3 |
|
5 |
2 |
4 |
|
6 |
2 |
0 |
19. Estimate the average value of \(f\) on the interval [0, 6].
20. Estimate the average value of \(g\) on the interval [0, 6].
In problems 21 – 26, find the average value of \(f\) on the given interval.
21. \(f(x)\) from the graph for \(0 \leq x \leq 2\).
22. \(f(x)\) from the graph for \(0 \leq x \leq 4\).
23. \(f(x)\) from the graph for \(1 \leq x \leq 6\).
24. \(f(x)\) from the graph for \(4 \leq x \leq 6\).
25. \(f(x) = 2x + 1\) for \(0 \leq x \leq 4\).
26. \(f(x) = x^2\) for \(0 \leq x \leq 2\).
The graph shows the velocity of a car during a 5 hour trip.
(a) Estimate how far the car traveled during the 5 hours.
(b) At what constant velocity should you drive in order to travel the same distance in 5 hours?
The graph shows the number of telephone calls per minute at a large company.
Estimate the average number of calls per minute
(a) From 8 am to 5 pm.
(b) From 9 am to 1 pm.