91Ó°ÊÓ

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\)-axis. $$y=\sqrt{x-4}, y=0, x=8$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\)-axis. $$y=e^{-x}, y=0, x=0, x=3$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\)-axis. $$y=\ln x, y=0, x=5$$

Use any method to find the arc length of the curve. $$y=2 x^{2}, 0 \leq x \leq 2$$

Use any method to find the arc length of the curve. $$y=3 \ln x, 1 \leq x \leq 3$$

Use any method to find the area of the surface generated by revolving the curve about the \(x\)-axis. $$y=\sin x, 0 \leq x \leq \pi$$

Use any method to find the area of the surface generated by revolving the curve about the \(x\)-axis. $$y=1 / x, 1 \leq x \leq 4$$

Information is given about the motion of a particle moving along a coordinate line. (a) Use a CAS to find the position function of the particle for \(t \geq 0 .\) You may approximate the constants of integration, where necessary. (b) Graph the position versus time curve. $$v(t)=20 \cos ^{6} t \sin ^{3} t, s(0)=2$$

Find a substitution that can be used to integrate rational functions of \(\sinh x\) and \(\cosh x\) and use your substitution to evaluate $$\int \frac{d x}{2 \cosh x+\sinh x}$$ without expressing the integrand in terms of \(e^{x}\) and \(e^{-x}.\)