91Ó°ÊÓ

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves. $$ x=y^{3}-y, x=0 $$

Find the mass and center of gravity of the lamina with density \(\delta .\) A lamina bounded by the \(x\) -axis, the line \(x=1\), and the curve \(y=\sqrt{x} ; \delta=2\).

Assume that \(y=f(x)\) is a smooth curve on the interval \([a, b]\) and assume that \(f(x) \geq 0\) for \(a \leq x \leq b\). Derive a formula for the surface area generated when the curve \(y=f(x), a \leq x \leq b\), is revolved about the line \(y=-k(k>0)\)

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ x=y^{2}, x=y+2 $$

The central span of the Golden Gate Bridge in California is \(4200 \mathrm{ft}\) long and is suspended from cables that rise \(500 \mathrm{ft}\) above the roadway on either side. Approximately how long is the portion of a cable that lies between the support towers on one side of the roadway? [Hint: As suggested by the accompanying figure, assume the cable is modeled by a parabola \(y=a x^{2}\) that passes through the point \((2100,500)\). Use a CAS or a calculating utility with a numerical integration capability to approximate the length of the cable. Round your answer to the nearest foot.]

Find \(d y / d x\). $$ y=\frac{1}{\tanh ^{-1} x} $$

A rocket weighing 3 tons is filled with 40 tons of liquid fuel. In the initial part of the flight, fuel is burned off at a constant rate of 2 tons per \(1000 \mathrm{ft}\) of vertical height. How much work in foot-tons (ft ton) is done lifting the rocket \(3000 \mathrm{ft}\) ?

Find \(d y / d x\). $$ y=\left(\operatorname{coth}^{-1} x\right)^{2} $$

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ x=1-y^{2}, x=2+y^{2}, y=-1, y=1 $$

Find the mass and center of gravity of the lamina with density \(\delta .\) A lamina bounded by the graph of \(x=y^{4}\) and the line \(x=1\); \(\delta=15\).