Chapter 9

In Problems \(47-50\), convert the given equation to rectangular coardinates. $$ \theta=\pi / 6 $$

Find the center of mass of the lamina that has the given shape and density. $$ y=\sqrt{9-x^{2}}, y=0 ; \rho(x, y)=x^{2} $$

Convert the given equation to rectangular cocrdinates. $$ \theta=\pi / 6 $$

Use (8) to find the indicated derivative. $$ z=u^{3} v-u v^{4} ; u=e^{-5 t}, v=\sec 5 t ; \frac{d z}{d t} $$

In Problems \(51-54\), find the volume of the solid that is bounded by the graphs of the given equations. $$ x^{2}+y^{2}=4, x^{2}+y^{2}+z^{2}=16, z=0 $$

Find the moment of inertia about the \(x\) -axis of the lamina that has the given shape and density. $$ x=y-y^{2}, x=0 ; \rho(x, y)=2 x $$

Use (8) to find the indicated derivative. $$ w=\cos (3 u+4 v) ; u=2 t+\frac{\pi}{2}, v=-t-\frac{\pi}{4},\left.\frac{d w}{d t}\right|_{t=\pi} $$

Use triple integrals and cylindrical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ x^{2}+y^{2}=4, \quad x^{2}+y^{2}+z^{2}=16, z=0 $$

In Problems \(51-54\), find the volume of the solid that is bounded by the graphs of the given equations. $$ z=10-x^{2}-y^{2}, \quad z=1 $$

Find the moment of inertia about the \(x\) -axis of the lamina that has the given shape and density. $$ y=x^{2}, y=\sqrt{x} ; \rho(x, y)=x^{2} $$

If \(\mathbf{v}\) is a constant vector and \(\mathbf{r}\) is integrable on \([a, b]\), prove that \(\int_{a}^{b} \mathbf{v} \cdot \mathbf{r}(t) d t=\mathbf{v} \cdot \int_{a}^{b} \mathbf{r}(t) d t\)

Use triple integrals and cylindrical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ z=10-x^{2}-y^{2}, z=1 $$

Use (8) to find the indicated derivative. $$ w=e^{x y} ; x=\frac{4}{2 t+1}, y=3 t+5 ;\left.\frac{d w}{d t}\right|_{t=0} $$

In Problems \(51-54\), find the volume of the solid that is bounded by the graphs of the given equations. $$ z=x^{2}+y^{2}, x^{2}+y^{2}=25, \quad z=0 $$

Find the moment of inertia about the \(x\) -axis of the lamina that has the given shape and density. $$ y=\cos x,-\pi / 2 \leq x \leq \pi / 2, y=0 ; \rho(x, y)=k \text { (constant) } $$

If \(u=f(x, y)\) and \(x=r \cos \theta, y=r \sin \theta\), show that Laplace's equation \(\partial^{2} u / \partial x^{2}+\partial^{2} u / \partial y^{2}=0\) becomes $$ \frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0 $$

Use triple integrals and cylindrical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ z=x^{2}+y^{2}, x^{2}+y^{2}=25, \quad z=0 $$

In Problems \(51-54\), find the volume of the solid that is bounded by the graphs of the given equations. $$ y=x^{2}+z^{2}, \quad 2 y=x^{2}+z^{2}+4 $$

Find the moment of inertia about the \(x\) -axis of the lamina that has the given shape and density. $$ y=\sqrt{4-x^{2}}, x=0, y=0, \text { first quadrant; } \rho(x, y)=y $$

The equation of state for a thermodynamic system is \(F(P, V, 7)=0\), where \(P, V\), and \(T\) are pressure, volume, and temperature, respectively. If the equation defines \(V\) as a function of \(P\) and \(T\), and also defines \(T\) as a function of \(V\) and \(P\), show that $$ \frac{\partial V}{\partial T}=-\frac{\frac{\partial F}{\partial T}}{\frac{\partial F}{\partial V}}=-\frac{1}{\frac{\partial T}{\partial V}} $$

Use triple integrals and cylindrical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ y=x^{2}+z^{2}, \quad 2 y=x^{2}+z^{2}+4 $$

Van der Waals' equation of state for the real gas \(\mathrm{CO}_{2}\) is $$ P=\frac{0.08 T}{V-0.0427}-\frac{3.6}{V^{2}} . $$ If \(d T / d t\) and \(d V / d t\) are rates at which the temperature and volume change, respectively, use the Chain Rule to find \(d P / d t\).

Find the centroid of the homogeneous solid that is bounded by the hemisphere \(z=\sqrt{a^{2}-x^{2}-y^{2}}\) and the plane \(z=0\)

Find the moment of inertia about the \(y\) -axis of the lamina that has the given shape and density. $$ y=x^{2}, x=0, y=4, \text { first quadrant; } \rho(x, y)=y $$

The equation of state for a thermodynamic system is \(F(P, V, 7)=0\), where \(P, V\), and \(T\) are pressure, volume, and temperature, respectively. If the equation defines \(V\) as a function of \(P\) and \(T\), and also defines \(T\) as a function of \(V\) and \(P\), show that $$ \frac{\partial V}{\partial T}=-\frac{\frac{\partial F}{\partial T}}{\frac{\partial F}{\partial V}}=-\frac{1}{\frac{\partial T}{\partial V}} . $$

Find the center of mass of the solid that is bounded by the graphs of \(y^{2}+z^{2}=16, x=0\), and \(x=5\) if the density at a point \(P\) is directly proportional to distance from the \(y z\) -plane.

The voltage across a conductor is increasing at a rate of 2 volts/min and the resistance is decreasing at a rate of 1 ohm/min. Use \(I=E / R\) and the Chain Rule to find the rate at which the current passing through the conductar is changing when \(R=50\) ohms and \(E=60\) volts.

Find the moment of inertia about the \(z\) -axis of the solid that is bounded above by the hemisphere \(z=\sqrt{9-x^{2}-y^{2}}\) and below by the plane \(z=2\) if the density at a point \(P\) is inversely proportional to the square of the distance from the \(z\) -axis.

Find the moment of inertia about the \(y\) -axis of the lamina that has the given shape and density. $$ y=x, y=0, y=1, x=3 ; \rho(x, y)=4 x+3 y $$

A particle moves in 3-space so that its coordinates at any time are \(x=4 \cos t, y=4 \sin t, z=5 t, t \geq 0\). Use the Chain Rule to find the rate at which its distance $$ w=\sqrt{x^{2}+y^{2}+z^{2}} $$ from the origin is changing at \(t=5 \pi / 2\) seconds.

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{2}{3}, \frac{\pi}{2}, \frac{\pi}{6}\right) $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{2}{3}, \frac{\pi}{2}, \frac{\pi}{6}\right) $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(5, \frac{5 \pi}{4}, \frac{2 \pi}{3}\right) $$

Find the radius of gyration about the indicated axis of the lamina that has the given shape and density. $$ x+y=a, a>0, x=0, y=0 ; \rho(x, y)=k \text { (constant); } x \text { -axis } $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(5, \frac{5 \pi}{4}, \frac{2 \pi}{3}\right) $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(8, \frac{\pi}{4}, \frac{3 \pi}{4}\right) $$

A lamina has the shape of the region bounded by the graph of the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). If its density is \(\rho(x, y)=1\), find: (a) the moment of inertia about the \(x\) -axis of the lamina, (b) the moment of inertia about the \(y\) -axis of the lamina, (c) the radius of gyration about the \(x\) -axis [Hint: The area of the ellipse is \(\pi a b]\), and (d) the radius of gyration about the \(y\) -axis.

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(8, \frac{\pi}{4}, \frac{3 \pi}{4}\right) $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{1}{3}, \frac{5 \pi}{3}, \frac{\pi}{6}\right) $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{1}{3}, \frac{5 \pi}{3}, \frac{\pi}{6}\right) $$

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ (-5,-5,0) $$

Find the polar moment of inertia of the lamina that has the given shape and density. $$ x+y=a, a>0, x=0, y=0 ; \rho(x, y)=k \text { (constant) } $$

Convert the points given in rectangular coordinates to spherical coordinates. $$ (-5,-5,0) $$

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ (1,-\sqrt{3}, 1) $$

Convert the points given in rectangular coordinates to spherical coordinates. $$ (1,-\sqrt{3}, 1) $$

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 1\right) $$

Find the polar moment of inertia of the lamina that has the given shape and density. \(x=y^{2}+2, x=6-y^{2}\); density at a point \(P\) inversely proportional to the square of the distance from the origin

Convert the points given in rectangular coordinates to spherical coordinates. $$ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 1\right) $$

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ \left(-\frac{\sqrt{3}}{2}, 0,-\frac{1}{2}\right) $$

Find the polar moment of inertia of the lamina that has the given shape and density. $$ y=x, y=0, y=3, x=4 ; \rho(x, y)=k \text { (constant) } $$