91Ó°ÊÓ

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ y=\cos \left(x^{2}\right), x=0, x=\frac{1}{2} \sqrt{\pi}, y=0 $$

Use a CAS to find the exact area of the surface generated by revolving the curve about the stated axis. $$ y=\sqrt{x}-\frac{1}{3} x^{3 / 2}, 1 \leq x \leq 3 ; x \text { -axis } $$

Find the volume of the solid whose base is the region bounded between the curve \(y=x^{2}\) and the \(x\) -axis from \(x=0\) to \(x=2\) and whose cross sections taken perpendicular to the \(x\) -axis are squares.

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ y=2 x-1, y=-2 x+3, x=2 $$

Assume that \(10 \mathrm{ft}\). Ib of work is required to stretch a spring \(1 \mathrm{ft}\) beyond its natural length. What is the spring constant?

Sketch the region enclosed by the curves and find its area. $$ y=\cos 2 x, y=0, x=\pi / 4, x=\pi / 2 $$

Determine whether the statement is true or false. Explain your answer. The graph of \(y=\sqrt{1-x^{2}}\) is a smooth curve on \([-1,1]\)

Suppose that a flat surface is immersed vertically in a fluid of weight density \(\rho .\) If \(\rho\) is doubled, is the force on the plate also doubled? Explain your reasoning.

Find \(d y / d x\) $$ y=\sinh (4 x-8) $$

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ y=2 x-x^{2}, y=0 $$