91Ó°ÊÓ

Suppose that \(a>b>0 .\) Let \(\varepsilon=\sqrt{a^{2}-b^{2}} / a\). Show that, for any \(0<\xi

For what a, \(0 \leq a \leq \pi,\) does the function \(f(x)=\sin (x)-\) \(\cos (x)\) have the greatest average over the interval \([a, a+\pi]\) of length \(\pi ?\)

Calculate the second moment of the given function \(f\) about the vertical axis \(x=c\) for the given \(c\). $$ f(x)=\sin (x) \quad 0 \leq x \leq \pi \quad c=2 \pi $$

In each of Exercises 57-60, use the method of disks to calculate the volume \(V\) of the solid obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=\sin (x), 0 \leq x \leq \pi\) and above the \(x\) -axis.

Calculate the variance \(\operatorname{Var}(X)\) of a random variable \(X\) whose probability density function is the given function \(f\). $$ f(x)=x / 4 \quad 1 \leq x \leq 3 $$

In each of Exercises \(58-61,\) calculate the area \(S\) of the surface obtained when the graph of the given function is rotated about the \(x\) -axis. $$ f(x)=\exp (x) \quad 0 \leq x \leq 1. $$

In each of Exercises 57-60, use the method of disks to calculate the volume \(V\) of the solid obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=\cos (x), \pi / 6 \leq x \leq \pi / 3\) and above the \(x\) -axis.

Given \(c>0\), for what value \(b, 0

In each of Exercises 57-60, use the method of disks to calculate the volume \(V\) of the solid obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the first quadrant region between the \(x\) -axis, the curve \(y=\sqrt{x} \cdot \exp (x),\) and the line \(x=1\).

Calculate the area \(S\) of the surface obtained when the graph of the given function is rotated about the \(x\) -axis. $$ f(x)=x^{4}+\frac{1}{32} x^{-2} \quad 1 / 2 \leq x \leq 1 $$