Chapter 5

Find the area under the curve \(y=1 / x^{3}\) from \(x=1\) to \(x=b\) and evaluate it for \(b=10,100,\) and \(1000 .\) Then find the total area under this curve for \(x \geqslant 1.\)

\(\begin{array}{l}{1-2 \text { Write the function as a sum of partial fractions. Do not }} \\ {\text { determine the numerical values of the coefficients. }} \\ {\text { 1. (a) } \frac{1}{x^{2}-1} \quad \text { (b) } \frac{2}{x^{2}+x}}\end{array}\)

Evaluate the integral. \(\int_{-2}^{3}\left(x^{2}-3\right) d x\)

Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\) \(\int x^{2} \ln x d x ; \quad u=\ln x, d v=x^{2} d x\)

Evaluate the Riemann sum for \(f(x)=3-\frac{1}{2} x\) \(2 \leqslant x \leqslant 14,\) with six subintervals, taking the sample points to be left endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

(a) Graph the functions \(f(x)=1 / x^{1.1}\) and \(g(x)=1 / x^{0.9}\) in the viewing rectangles \([0,10]\) by \([0,1]\) and \([0,100]\) by \([0,1] .\) (b) Find the areas under the graphs of \(f\) and \(g\) from \(x=1\) to \(x=b\) and evaluate for \(b=10,100,10^{4}, 10^{6}, 10^{10},\) and \(10^{20}\) . (c) Find the total area under each curve for \(x \geqslant 1,\) if it exists.

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. \(\int e^{2 \theta} \sin 3 \theta d \theta\)

\(1-2\) Write the function as a sum of partial fractions. Do not determine the numerical values of the coefficients. $$\frac{x}{x^{2}+x-2} \quad \text { (b) } \frac{2-x}{x^{2}-2 x-8}$$

Evaluate the integral. \(\int_{1}^{2} x^{-2} d x\)

Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\) \(\int \theta \cos \theta d \theta ; \quad u=\theta, d v=\cos \theta d \theta\)

Evaluate the integral by making the given substitution. \(\int x^{3}\left(2+x^{4}\right)^{5} d x, u=2+x^{4}\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{3}^{\infty} \frac{1}{(x-2)^{3 / 2}} d x\)

3-14 Evaluate the integral. $$\int \frac{x}{x-6} d x$$

Evaluate the integral. \(\int_{0}^{2}\left(x^{4}-\frac{3}{4} x^{2}+\frac{2}{3} x-1\right) d x\)

Evaluate the integral. \(\int x \cos 5 x d x\)

3\. a) Estimate the area under the graph of \(f(x)=\cos x\) from \(x=0\) to \(x=\pi / 2\) using four approximating rectangles and right endpoints. Sketch the graph and the rect- angles. Is your estimate an underestimate or an overestimate? b) Repeat part (a) using left endpoints.

Evaluate the integral by making the given substitution. \(\int x^{2} \sqrt{x^{3}+1} d x, u=x^{3}+1\)

If $$f(x)=e^{x}-2,0 \leq x \leqslant 2$$, find the Riemann sum with \(n=4\) correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{0}^{\infty} \frac{1}{\sqrt[4]{1+x}} d x\)

Evaluate the integral. $$\int \frac{r^{2}}{r+4} d r$$

Evaluate the integral. \(\int_{0}^{1}\left(1+\frac{1}{2} u^{4}-\frac{2}{5} u^{9}\right) d u\)

Evaluate the integral. \(\int x e^{-x} d x\)

4\. (a) Estimate the area under the graph of \(f(x)=\sqrt{x}\) from \(x=0\) to \(x=4\) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

Evaluate the integral by making the given substitution. \(\int \frac{d t}{(1-6 t)^{4}}, \quad u=1-6 t\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{-1} \frac{1}{\sqrt{2-w}} d w\)

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. \(\int e^{2 x} \arctan \left(e^{x}\right) d x\)

Evaluate the integral. $$\int \frac{x-9}{(x+5)(x-2)} d x$$

Evaluate the integral. \(\int_{0}^{1} x^{4 / 5} d x\)

Evaluate the integral. \(\int r e^{r / 2} d r\)

5\. (a) Estmate the area under the graph of \(f(x)=1+x^{2}\) from \(x=-1\) to \(x=2\) using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (b) Repeat part (a) using midpoints. (d) From your sketches in parts (a) - (c), which appears to be the best estimate?

Evaluate the integral by making the given substitution. \(\int \cos ^{3} \theta \sin \theta d \theta, \quad u=\cos \theta\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{0}^{\infty} \frac{x}{\left(x^{2}+2\right)^{2}} d x\)

Evaluate the integral. $$\int \frac{1}{(t+4)(t-1)} d t$$

6\. (a) Graph the function \(f(x)=x-2 \ln x \quad 1 \leqslant x \leq 5\) (b) Estimate the area under the graph of \(f\) using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles.

Evaluate the integral. \(\int_{1}^{8} \sqrt[3]{x} d x\)

Evaluate the integral. \(\int t \sin 2 t d t\)

Evaluate the integral by making the given substitution. \(\int \frac{\sec ^{2}(1 / x)}{x^{2}} d x, \quad u=1 / x\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{4}^{\infty} e^{-y / 2} d y\)

Evaluate the integral. $$\int_{2}^{3} \frac{1}{x^{2}-1} d x$$

7\. The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline t(s) & {0} & {0.5} & {1.0} & {1.5} & {2.0} & {2.5} & {3.0} \\ \hline v(f t / s) & {0} & {6.2} & {10.8} & {14.9} & {18.1} & {19.4} & {20.2} \\ \hline\end{array}$$

Evaluate the integral. \(\int_{-1}^{0}\left(2 x-e^{x}\right) d x\)

Evaluate the integral. \(\int x^{2} \sin \pi x d x\)

Evaluate the indefinite integral. \(\int x \sin \left(x^{2}\right) d x\)

A table of values of an increasing function \(f\) is shown. Use the table to find lower and upper estimates for \(\int_{10}^{30} f(x) d x\) $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {10} & {14} & {18} & {22} & {26} & {30} \\ \hline f(x) & {-12} & {-6} & {-2} & {1} & {3} & {8} \\\ \hline\end{array}$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{-1} e^{-2 t} d t\)

Evaluate the integral. $$\int_{0}^{1} \frac{x-1}{x^{2}+3 x+2} d x$$

8\. Speedometer readings for a motorcycle at 12 -second intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\\ \hline\end{array}$$

Evaluate the integral. \(\int_{-5}^{5} e d x\)

Evaluate the integral. \(\int x^{2} \cos m x d x\)

Evaluate the indefinite integral. \(\int x^{2}\left(x^{3}+5\right)^{9} d x\)