MA005: Calculus I

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
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