Practice Problems

In problems 1 – 3, use the values in Table 1 to estimate the areas.

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
This work is licensed under a Creative Commons Attribution 3.0 License.

1. between 11 (using left endpoints of intervals) and 6 (using right endpoints)

3. between 4 (using left endpoints of intervals) and 6 (using right endpoints)

5. Using left endpoint widths: $(0)(40)+(70)(40)+(55)(40)+(90)(40)+(130)(40)+(115)(40)=18,400 \mathrm{ft}^{2}$

Right endpoint widths $(70,55, \ldots)$ and average widths $(70 / 2,125 / 2, \ldots)$ give the same result $18,400 \mathrm{ft}^{2}$. All of these are reasonable methods for estimating the area of the island.

7. $9$

9. $1$

11. $\frac{1}{2} \cdot \mathrm{e}^{2}-\frac{3}{2}$

13. $\frac{1}{32} \pi^{2}+\frac{1}{4} \pi-\frac{\sqrt{2}}{2}$

15. $e^{2}-3$

17. $3-\frac{\pi}{4}$

19. Estimate using midpoints of unit intervals: $\frac{1}{4}\{\mathrm{f}(1)(1)+\mathrm{f}(2)(1)+\mathrm{f}(3)(1)+\mathrm{f}(4)(1)\}=\frac{19}{4}$. About $\frac{19}{4}$.

21. Estimate using midpoints of unit intervals: $\frac{1}{2}\{\mathrm{f}(2)(1)+\mathrm{f}(3)(1)\}=5$. About 5.

23. $\text { average } \approx 1$

25. $\text { average } \approx \frac{11}{5}$

27. $\text { average }=5$

29. $\text { average }=\frac{13}{3}$

31. $\text { average }=\frac{2}{\pi}$

33.

(a) $\mathrm{C}=1: \text { average }=\frac{2}{3}$

(b) $\mathrm{C}=9: \text { average }=2$

(c) $\mathrm{C}=81: \text { average }=6$

(d) $\mathrm{C}=100: \text { average }=\frac{20}{3}$

In general, $\text { average }=\frac{2}{3} \sqrt{\mathrm{C}}$.

35.

(a) Graphically, $\text { average } \approx 3000 \cdot 1000 \frac{\text { calls }}{\text { hour }}=\frac{3000000}{60} \frac{\text { calls }}{\mathrm{min}} \approx 50,000 \frac{\text { calls }}{\mathrm{min}}$

(b) About $58,333 \frac{\text { calls }}{\mathrm{min}}$

37.

(a) Similar to Example 5: $\text { work }=1,950 \text { foot-pounds }$

(b) $\text { work }=1,312.5 \text { foot-pounds }$

39.

(a) $\text { work } = 1,200 foot–pounds$

(b) $\text { work }=600 \text { foot-pounds }$

(c) $\text { work }=400 \text { foot-pounds }$

41. $\text { work }=1,275 \text { foot-pounds }$