Practice Problems

1. Calculate the areas of the shaded regions in Fig. 20.

3. Break Fig. 22 into a triangle and rectangle and verify that the total area of the trapezoid is $\mathrm{b} \cdot\left(\frac{\mathrm{h}+\mathrm{H}}{2}\right)$.

5. 

(a) Calculate the sum of the rectangular areas in Fig. 24a.
(b) From part (a), what can we say about the area of the shaded region in Fig. 24b?

7.Let $\mathrm{A}(x)$ represent the area bounded by the graph and the horizontal axis and vertical lines at $t=0$ and $t=x$ for the graph in Fig. 25. Evaluate $\mathrm{A}(x)$ for $x=1,2,3,4 \text {, and } 5 \text {. }$

9. Let $\mathrm{C}(x)$ represent the area bounded by the graph and the horizontal axis and vertical lines at $t=0$ and $t=x$ for the graph in Fig. 27. Evaluate  $\mathrm{B}(x)$ for $\mathrm{C}(x) \text { for } x=1,2, \text { and } 3$ and find a formula for $\mathrm{C}(x)$.

11. A car had the velocity given in Fig. 29. How far did the car travel from $\mathrm{t}=0 \text { to } \mathrm{t}=30 \text { seconds? }$

13. The velocities of two cars are shown in Fig. 31. (a) From the time the brakes were applied, how many seconds did it take each car to stop? (b) From the time the brakes were applied, which car traveled farther until it came to a complete stop?

15. What are the units for the "area" of a rectangle with the given base and height units?

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.1-Introduction-to-Integration.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. (a) $(10)(12) + (8)(4) = 152$  (b)$(10)(20) – (3)(8) = 176$

3. $\mathrm{bh}+\frac{1}{2} \mathrm{~b}(\mathrm{H}-\mathrm{h})=\mathrm{bh}+\frac{1}{2} \mathrm{bH}-\frac{1}{2} \mathrm{bh}=\mathrm{b}\left(\frac{\mathrm{h}+\mathrm{H}}{2}\right)$

5. (a) $(1)(3) + (1)(2) = 5$  (b) {area of shaded region in Fig. 24b} < 5

7. $\mathrm{A}(1)=1, \mathrm{~A}(2)=2.5, \mathrm{~A}(3)=4.5, \mathrm{~A}(4)=6, \mathrm{~A}(5)=7$

9. $\mathrm{C}(1)=1.5, \mathrm{C}(2)=4, \mathrm{C}(3)=7.5$ and $\mathrm{C}(\mathrm{x})=\text { rect. }+\text { triangle areas }=\mathrm{x}+\frac{1}{2} \mathrm{x}^{*} \mathrm{x}=\mathrm{x}+\frac{1}{2} \mathrm{x}^{2}$

11. Distance = "area" = $(20)(30)+\frac{1}{2}(10)(30)=600+150=750 \text { feet. }$

13. (a) A: 20 seconds to stop. B: 40 seconds to stop.
      (b) A: $\text { A: } \frac{1}{2}(20)(80)$ feet to stop B: $\frac{1}{2}(40)(40)=800$ feet to stop

15. miles, dollars, cubic feet, kilowatt. hours, people, square meals