91Ó°ÊÓ

Calculate the variance \(\operatorname{Var}(X)\) of a random variable \(X\) whose probability density function is the given function \(f\). $$ f(x)=3 x^{2} \quad 0 \leq x \leq 1 $$

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 between the \(x\) -axis, the curve \(y=x \sqrt{\sin (x)}, 0 \leq x \leq \pi\)

Calculate the variance \(\operatorname{Var}(X)\) of a random variable \(X\) whose probability density function is the given function \(f\). $$ f(x)=8 /\left(3 x^{3}\right) \quad 1 \leq x \leq 2 $$

Calculate the area \(S\) of the surface obtained when the graph of the given function is rotated about the \(x\) -axis. $$ f(x)=\sin (x)^{32} \quad 0 \leq x \leq \pi / 2 $$

Calculate the area \(S\) of the surface obtained when the graph of the given function is rotated about the \(x\) -axis. $$ f(x)=\frac{3 x^{4}+1}{6 x} \quad 1 \leq x \leq 2 $$

In each of Exercises 61-64, use the method of disks to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region between the curves \(y=\exp (x)\), the \(y\) -axis, and the line \(y=e\).

In each of Exercises 61-64, use the method of disks to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the first quadrant region between the curve \(y=\) \(\sin \left(x^{2}\right),\) the \(y\) -axis, and the line \(y=1\)

A population \(P\) satisfies the differential equation $$ P^{\prime}(t)=10^{-5} \cdot P(t) \cdot(15000-P(t)) $$ For what value \(P(0)\) of the initial population is the initial growth rate \(P^{\prime}(0)\) greatest?

A \(1 \mathrm{~kg}\) object is dropped. After 1 second, its (downward) speed is \(9.4 \mathrm{~m} / \mathrm{s}\). Assuming that the drag is proportional to the velocity, what is the terminal velocity of the object?

In each of Exercises 61-64, use the method of disks to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region between the curve \(y=x^{2} /\left(1-x^{2}\right),\) the \(y\) -axis, and the line \(y=1 / 3\).