91Ó°ÊÓ

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=\ln (\sin (x)) \quad I=[\pi / 4, \pi / 2] $$

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 \(y=\sqrt{x}\) below by the \(x\) -axis, and on the right by \(x=4\).

Solve the given differential equation. $$ \left(4+y^{2}\right) \frac{d y}{d x}=x^{2} $$

In each of Exercises 7-12, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region that is bounded on the left by the \(y\) -axis, on the right by the curve \(y=x^{2}, 1 \leq x \leq 2,\) and that is between the horizontal lines \(y=1\) and \(y=4\).

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 region bounded above by \(y=1 / x\), below by the \(x\) axis, and on the sides by the vertical lines \(x=1\) and \(x=2\).

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

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=\ln (x) \quad I=[1, e] $$

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=\frac{2}{3}\left(x^{2}+1\right)^{3 / 2} \quad I=[0,1] $$

In each of Exercises 7-12, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region bounded above by \(y=2,\) below by \(y=\sqrt{x}\) and on the left by the \(y\) -axis.

Solve the given differential equation. $$ \frac{d y}{d x}+3 x y^{2}=0 $$