Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
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Date: Tuesday, July 23, 2024, 5:24 AM

Description

Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

Table of contents

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

Answers

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.