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. $$y=\sqrt{x}, x=4, x=9, y=0$$

A spring exerts a force of \(100 \mathrm{N}\) when it is stretched \(0.2 \mathrm{m}\) beyond its natural length. How much work is required to stretch the spring \(0.8 \mathrm{m}\) beyond its natural length?

Sketch the region enclosed by the curves and find its area. $$y=x^{2}, y=\sqrt{x}, x=\frac{1}{4}, x=1$$

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. $$y=1 / x, y=0, x=1, x=3$$

Find the area of the surface generated by revolving the given curve about the \(y\) -axis. $$x=\sqrt{9-y^{2}},-2 \leq y \leq 2$$

Sketch the region enclosed by the curves and find its area. $$y=x^{3}-4 x, y=0, x=0, x=2$$

Find the area of the surface generated by revolving the given curve about the \(y\) -axis. $$x=2 \sqrt{1-y},-1 \leq y \leq 0$$

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. $$y=\cos \left(x^{2}\right), x=0, x=\frac{1}{2} \sqrt{\pi}, y=0$$

A spring whose natural length is \(15 \mathrm{cm}\) exerts a force of \(45 \mathrm{N}\) when stretched to a length of \(20 \mathrm{cm} .\) (a) Find the spring constant (in newtons/meter). (b) Find the work that is done in stretching the spring \(3 \mathrm{cm}\) beyond its natural length. (c) Find the work done in stretching the spring from a length of \(20 \mathrm{cm}\) to a length of \(25 \mathrm{cm} .\)

Find the exact arc length of the curve over the interval. $$x=\frac{1}{8} y^{4}+\frac{1}{4} y^{-2} \text {from } y=1 \text { to } y=4$$