91Ó°ÊÓ

Use the method of cylindrical shells to ind the volume generated by rotating the region bounded by the curves about the given axis. \(y=e^{-x}, y=0, x=-1, x=0 ;\) about \(x=1\)

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

Evaluate the integral. $$ \int \frac{1}{\sqrt{\sqrt{x}+1}} d x $$

Find the area of the region under the given curve from 1 to 2. $$ y=\frac{1}{x^{3}+x} $$

Use the method of cylindrical shells to ind the volume generated by rotating the region bounded by the curves about the given axis. \(y=e^{x}, x=0, y=3 ;\) about the \(x\) -axis

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

Find the area of the region under the given curve from 1 to 2. $$ y=\frac{x^{2}+1}{3 x-x^{2}} $$

$$ \begin{array}{l}{\text { A particle moves on a straight line with velocity function }} \\ {v(t)=\sin \omega t \cos ^{2} \omega t . \text { Find its position function } s=f(t) \text { if }} \\ {f(0)=0}\end{array} $$

Find the escape velocity \(v_{0}\) that is needed to propel a rocket of mass \(m\) out of the gravitational field of a planet with mass \(M\) and radius \(R\). Use Newton's Law of Gravitation (see Exercise 6.4 .33 ) and the fact that the initial kinetic energy of \(\frac{1}{2} m v_{0}^{2}\) supplies the needed work.

Calculate the volume generated by rotating the region bounded by the curves \(y=\ln x, y=0,\) and \(x=2\) about each axis. $$ \text { (a) The } y \text { -axis } \quad \text { (b) The } x \text { -axis } $$