Chapter 5: Multiple Integrals

For the cone in Problem 18, find ${\mathbf{l}}_{\mathbf{x}}\mathbf{/}\mathbf{M}\mathbf{,}\mathbf{}{\mathbf{l}}_{\mathbf{y}}\mathbf{}\mathbf{/}\mathbf{M}\mathbf{,}\mathbf{}{\mathbf{l}}_{\mathbf{z}}\mathbf{/}\mathbf{M}$. Also find $\mathbf{l}\mathbf{/}\mathbf{M}$ about a line through the center of mass parallel to the x axis.

In the integral

$\mathbf{I}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}{{\left(x^2+y^2\right)}^{\mathbf{2}}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{xy}}\mathbf{dxdy}$.

Make the change of variables

$$\begin{gathered} \mathbf{u}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}} \\ \mathbf{v}\mathbf{=}\mathbf{2}\mathbf{xy} \end{gathered}$$

And evaluate I. Hint: Use (4.8) and the accompanying discussion.

Above the square with vertices at, (0,0), (2,0),(0,2) and (2,2) and under the plane z = 8-x+y.

In Problems 17 to 30, for the curve $\mathit{y}\mathbf{=}\sqrt{\mathbf{x}}$, between $\mathit{x}\mathbf{=}\mathbf{0}$and $\mathit{x}\mathbf{=}\mathbf{2}$ , find:

The volume of the solid generated when the area is revolved about the axis.

Verify each of the following answers for an indefinite integral by one or more of the methods suggested above.

1. $\mathbf{\int }\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{d}\mathit{\theta }\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\theta }$or$\mathbf{-}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }$or$\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{\theta }$

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{2}}^{\mathbf{4}}\mathbf{3}\mathbf{xdydx}$

Prove the “parallel axis theorem”: The moment of inertia of a body about a given axis is $\mathbf{l}\mathbf{=}{\mathbf{l}}_{\mathbf{m}}\mathbf{+}{\mathbf{Md}}^{\mathbf{2}}$, where M is the mass of the body,${\mathrm{l}}_{\mathrm{m}}$is the moment of inertia of the body about an axis through the center of mass and parallel to the given axis, and dis the distance between the two axes.

Find the area of the plane $\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{5}\mathbf{z}\mathbf{=}\mathbf{13}$ cut out by the cylinder ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{9}$.

As needed, use a computer to plot graphs of figures and to check values of integrals

For the disk$\mathit{r}\mathbf{\le }\mathit{a}$, find by integration using polar coordinates:

(a) The area of the disk;

(b) The centroid of one quadrant of the disk;

(c) The moment of inertia of the disk about a diameter;

(d) The circumference of the circle$\mathit{r}\mathbf{=}\mathit{a}$;

(e) The centroid of a quarter circle arc.

(a) Find the area of the surface $\mathbf{z}\mathbf{=}\mathbf{1}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}$inside the cylinder ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$

(b) Find the volume inside the cylinder between the surface and the $\left(x,y\right)$ plane. Use cylindrical coordinates