Chapter 8

Evaluate the following integrals. $$\int \frac{d x}{1-x^{2}+\sqrt{1-x^{2}}}$$

Volumes of solids Find the volume of the following solids. The region bounded by $$y=\frac{x}{x+1},\( the \)x\( -axis, and \)x=4$$ is revolved about the \(x\) -axis.

Comparing volumes Let \(R\) be the region bounded by \(y=\sin x\) and the \(x\) -axis on the interval \([0, \pi]\). Which is greater, the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or the volume of the solid generated when \(R\) is revolved about the \(y\) -axis?

Different substitutions a. Evaluate \(\int \tan x \sec ^{2} x d x\) using the substitution \(u=\tan x\) b. Evaluate \(\int \tan x \sec ^{2} x d x\) using the substitution \(u=\sec x\) c. Reconcile the results in parts (a) and (b).

A useful integral a. Use integration by parts to show that if \(f^{\prime}\) is continuous, then $$\int x f^{\prime}(x) d x=x f(x)-\int f(x) d x$$. b. Use part (a) to evaluate \(\int x e^{3 x} d x\).

Use a computer algebra system to solve the following problems. Find the approximate area of the surface generated when the curve \(y=1+\sin x+\cos x,\) for \(0 \leq x \leq \pi,\) is revolved about the \(x\) -axis.

Evaluate the following integrals. $$\int \ln \left(x^{2}+a^{2}\right) d x, a \neq 0$$

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=\frac{\sqrt{x}}{\sqrt[3]{x^{2}+1}}\) and the \(x\) -axis on the interval \([0, \infty)\) is revolved about the \(x\) -axis.

Volumes of solids Find the volume of the following solids. The region bounded by $$y=\frac{1}{x^{2}\left(x^{2}+2\right)^{2}}, y=0, x=1,$$ and \(x=2\) is revolved about the \(y\) -axis.

Using the integral of sec \(^{3} u\) By reduction formula 4 in Section 8.3 $$ \int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C $$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(4+x^{2}\right)^{1 / 2},[0,2]$$