Practice Problems

In problems 1 – 3 , rewrite the limit of each Riemann sum as a definite integral.

1. $\lim _{\text {mesh } \rightarrow 0}\left(\sum_{k=1}^{n}\left(2+3 c_{k}\right) \Delta x_{k}\right)$ on the interval $[ 0, 4]$

3. $\lim _{\operatorname{mesh} \rightarrow 0}\left(\sum_{k-1}^{n}\left(c_{k}\right)^{3} \Delta x_{k}\right)$ on $[ 2, 5]$

In problems 5 – 9, represent the area of each bounded region as a definite integral. (Do not evaluate the integral, just translate the area into an integral.)

5. The region bounded by $\mathrm{y}=x^{3}$, the x–axis, the line $x = 1$, and $x = 5$.

7.The region bounded by $y=x \cdot \sin (x)$, the x–axis, the line $x=1 / 2$ , and $x=2$.

9. The shaded region in Fig. 10.

In problems 11 – 15 , represent the area of each bounded region as a definite integral, and use geometry to determine the value of the definite integral.

11. The region bounded by the x–axis, the line $x = 1$, and $x = 3$.

13. The region bounded by $\mathrm{y}=\mathbf{I} x \mathrm{I}$, the x–axis, and the line $x = –1$.

15. The shaded region in Fig. 12.

17. Fig. 14 shows the graph of $g$ and the areas of several regions.

Evaluate:

(a) $\int_{1}^{3} \mathrm{~g}(x) \mathrm{dx}$

(b) $\int_{3}^{4} \mathrm{~g}(x) \mathrm{dx}$

(c) $\int_{4}^{8} \mathrm{~g}(x) \mathrm{dx}$

(d) $\int_{1}^{8} g(x) d x$

(e) $\int_{3}^{8}|g(x)| d x$

In problem 19 , your velocity (in feet per minute) along a straight path is shown. (a) Sketch the graph of your location. (b) How many feet did you walk in 8 minutes? (c) Where, relative to your starting location, are you after 8 minutes?

19. Your velocity is shown in Fig. 16.

In problems 21 – 27, the units are given for $x$ and $\text { a } \mathrm{f}(x)$. Give the units of $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} \mathrm{x}$.

21. $x$ is time in "seconds", and $\mathrm{f}(x)$ is velocity in "meters per second".

23. $x$ is a position in "feet", and $\mathrm{f}(x)$ is an area in "square feet".

25. $x$ is a height in "meters", and $\mathrm{f}(x)$ is a force in "grams".

27. $x$ is a time in "seconds", and $\mathrm{f}(x)$ is an acceleration in "feet per second per second $\left(\mathrm{ft} / \mathrm{s}^{2}\right)$.

29. For $f(x)=x^{3}$, partition the interval [0,2] into n equally wide subintervals of length $\Delta \mathrm{x}=2 / \mathrm{n}$. Write the lower sum for this function and partition, and calculate the limit of the lower sum as $\mathrm{n} \rightarrow \infty$. (b) Write the upper sum for this function and partition and find the limit of the upper sum as $\mathrm{n} \rightarrow \infty$.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.3-Definite-Integral.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. $\int_{0}^{4} 2+3 x d x$

3. $\int_{2}^{5} x^{3} d x$

5. $\int_{1}^{5} x^{3} d x$

7. $\int_{0.5}^{2} x \sin (x) d x$

9. $\int_{1}^{3} \ln (x) d x$

11. $\int_{1}^{3} 2 x d x=8$

13. $\int_{-1}^{0}|x| \mathrm{dx}=1 / 2$

15. $\begin{aligned} &\int_{0}^{4} 3-\frac{x}{2} d x=8 \\ \end{aligned}$

17. (a) $3$ (b) $–1$ (c) $6$ (d) 8 (e) $7$

19.
(a) see the graph
(b) 24 feet.
(c) 24 feet from the starting point.

21. meters

23. feet³ = cubic feet

25. gram.meters

27. feet/second = feet per second

29. $\Delta x=\frac{2-0}{n}=\frac{2}{n} \cdot m_{i}=\frac{2}{n}(i-1) \text { and } M_{i}=\frac{2}{n} i \text { so } f\left(m_{i}\right)=\left\{\frac{2}{n}(i-1)\right\}^{3} \text { and } f\left(M_{i}\right)=\left\{\frac{2}{n} i\right\}^{3}$

a. $\mathrm{LS}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{m}_{\mathrm{i}}\right) \Delta \mathrm{x}=\sum_{\mathrm{i}=1}^{\mathrm{n}}\left\{\frac{2}{\mathrm{n}}^{(\mathrm{i}-1)}\right\}^{3} \frac{2}{\mathrm{n}}=\frac{2}{\mathrm{n}} \frac{8}{\mathrm{n}^{3}}\left\{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{i}^{3}-3 \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{i}^{2}+3 \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{i}-\sum_{\mathrm{i}=1}^{\mathrm{n}} 1\right\}$

$=\frac{16}{n^{4}}\left\{\left(\frac{1}{4} n^{4}+\frac{1}{2} n^{3}+\frac{3}{12} n^{2}\right)-3\left(\frac{1}{3} n^{3}+\frac{1}{2} n^{2}+\frac{2}{12} n\right)+3\left(\frac{1}{2} n^{2}+\frac{1}{2} n\right)-n\right\}$

$=\frac{16}{n^{4}}\left\{\frac{1}{4} n^{4}-\frac{1}{2} n^{3}+\frac{1}{4} n^{2}\right\}=4-\frac{8}{n}+\frac{4}{n^{2}} \longrightarrow 4$.

b. $\mathrm{US}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{M}_{\mathrm{i}}\right) \Delta \mathrm{x}=\sum_{\mathrm{i}=1}^{\mathrm{n}}\left\{\frac{2}{\mathrm{n}} \mathrm{i}\right\}^{3} \frac{2}{\mathrm{n}}=\frac{2}{\mathrm{n}} \frac{8}{\mathrm{n}^{3}}\left\{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{i}^{3}\right\}=\frac{16}{\mathrm{n}}\left\{\frac{1}{4} \mathrm{n}^{4}+\frac{1}{2} \mathrm{n}^{3}+\frac{3}{12} \mathrm{n}^{2}\right\}$

$=4+\frac{8}{n}+\frac{4}{n^{2}} \longrightarrow 4$.