91Ó°ÊÓ

Solve the given differential equation. $$ x \frac{d y}{d x}-y=3 x^{3} y $$

Calculate the arc length of the graph of the given equation. $$ x=(y-8)^{3 / 2}+2 \quad 8 \leq y \leq 12 $$

Find the moment of the given region \(\mathcal{R}\) about the \(x\) -axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the first quadrant region bounded above by \(y=1-x^{3}\) and below by the \(x\) -axis.

Calculate the arc length of the graph of the given equation. $$ x^{2} / y^{3}=4 \quad 1 / 3 \leq y \leq 7 $$

In each of Exercises 13-18, use the method of washers to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region bounded above by the curve \(y=4-x^{2}\) and below by the line \(y=x+2\).

Solve the given differential equation. $$ \exp (2 x+3 y) \frac{d y}{d x}=1 $$

In each of Exercises \(13-16,\) a function \(f\) and an interval \(I=[a, b]\) are given. Find a number \(c\) in \((a, b)\) for which \(f(c)\) is the average value of \(f\) on \(I\). $$ f(x)=x^{2}-10 x / 3 \quad I=[1,3] $$

Find the moment of the given region \(\mathcal{R}\) about the \(x\) -axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the first quadrant region bounded above by \(y=6 x /\left(1+x^{3}\right),\) below by the \(x\) -axis, and on the right by \(x=1\)

In each of Exercises 13-18, use the method of washers to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region bounded above by the line \(y=2\) and below by the curve \(y=x^{2}+1\)

In each of Exercises \(13-16,\) a function \(f\) and an interval \(I=[a, b]\) are given. Find a number \(c\) in \((a, b)\) for which \(f(c)\) is the average value of \(f\) on \(I\). $$ f(x)=\ln (x) \quad I=[1, e] $$