91Ó°ÊÓ

Find the volume obtained by rotating the region bounded by the given curves about the specified axis. $$y=\sin ^{2} x, y=0,0 \leqslant x \leqslant \pi ; \quad \text { about the } x$$

\(1-80\) Evaluate the integral. $$\int \frac{1}{x+\sqrt[3]{x}} d x$$

The average speed of molecules in an ideal gas is $$\overline{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2}\int_{0}^{\infty} v^{3} e^{-M s^{2} / 2 R T_{1}} d v$$ where \(M\) is the molecular weight of the gas, \(R\) is the gas con- stant, \(T\) is the gas temperature, and \(v\) is the molecular speed. Show that $$\overline{v}=\sqrt{\frac{8 R T}{\pi M}}$$

\(\begin{array}{l}{62-63 \text { Find the area of the region under the given curve from }} \\ {1 \text { to } 2 \text { . }}\end{array}\) $$ y=\frac{x^{2}+1}{3 x-x^{2}} $$

A particle that moves along a straight line has velocity \(v(t)=t^{2} e^{-t}\) meters per second after \(t\) seconds. How far will it travel during the first \(t\) seconds?

We know from Example 1 that the region \(\mathscr{R}=\\{(x, y) | x \geqslant 1,0 \leq y \leq 1 / x\\}\) has infinite area. Show that by rotating \(\mathscr{T}\) about the \(x\) -axis we obtain a solid with finite volume.

Find the volume obtained by rotating the region bounded by the given curves about the specified axis. $$y=\sin x, y=\cos x, 0 \leqslant x \leqslant \pi / 4 ; \quad \text { about } y=1$$

\(1-80\) Evaluate the integral. $$\int \frac{\sin 2 x}{1+\cos ^{4} x} d x$$

\(1-80\) Evaluate the integral. $$\int_{\pi / 4}^{\pi / 3} \frac{\ln (\tan x)}{\sin x \cos x} d x$$

Find the volume obtained by rotating the region bounded by the given curves about the specified axis. $$y=\sec x, y=\cos x, 0 \leqslant x \leqslant \pi / 3 ; \quad \text { about } y=-1$$