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
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