91Ó°ÊÓ

Find the area of the region enclosed by the curves \(y=x^{2}\) and \(y=4 x\) by integrating (a) with respect to \(x\) (b) with respect to \(y\)

In each part, a value for one of the hyperbolic functions is given at an unspecified positive number \(x_{0} .\) Use appropriate identities to find the exact values of the remaining five hyperbolic functions at \(x_{0}\) (a) \(\sinh x_{0}=2\) (b) \(\cosh x_{0}=\frac{5}{4}\) (c) \(\tanh x_{0}=\frac{4}{3}\)

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?

Find the exact arc length of the curve over the stated interval. $$y=x^{2 / 3} \text { from } x=1\text { to } x=8$$

Find the exact arc length of the curve over the stated interval. $$y=\left(x^{6}+8\right) / 16 x^{2} \text { from } x=2 \text { to } x=3$$

Find the area of the region enclosed by the curves \(y^{2}=4 x\) and \(y=2 x-4\) by integrating (a) with respect to \(x\) (b) with respect to \(y\)

Obtain the derivative formulas for esch \(x,\) sech \(x,\) and \(\operatorname{coth} x\) from the derivative formulas for \(\sinh x, \cosh x,\) and \(\tanh x\)

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

Assume that a force of \(6 \mathrm{N}\) is required to compress a spring from a natural length of \(4 \mathrm{m}\) to a length of \(3 \frac{1}{2} \mathrm{m} .\) Find the work required to compress the spring from its natural length to a length of \(2 \mathrm{m}\). (Hooke's law applies to compression as well as extension.)

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