Chapter 5: Multiple Integrals

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 moments of a thin shell whose shape is the curved surface of the solid (assuming constant density).

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{4}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\sqrt{\mathbf{x}}}\sqrt[\mathbf{y}]{\mathbf{x}}\mathbf{dydx}$

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 mass of a wire bent in the shape of the arc if its density (mass per unit length) is.

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}}}\mathbf{ydxdy}$

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

The mass of the solid of revolution if the density (mass per unit volume) is $\left|\mathrm{xyz}\right|$.

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{\tau \tau }}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{y}}^{\mathbf{\tau \tau }}\mathbf{dxdy}$

Find the surface area cut from the cone$\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{5}{\mathbf{z}}^{\mathbf{2}}\mathbf{,}\mathbf{z}\mathbf{>}\mathbf{0}$by the cylinder${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{y}$

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

Using polar coordinates:

a. Show that the equation of the circle sketched is $\mathit{r}\mathbf{=}\mathbf{2}\mathbf{}\mathit{a}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$. Hint: Use the right triangleOPQ.

b. By integration, find the area of the disk$\mathit{r}\mathbf{\le }\mathbf{2}\mathbf{}\mathit{a}\mathbf{}\mathbf{cos}\mathbf{}\mathit{\theta }$.

c. Find the centroid of the area of the first quadrant half disk

d. Find the moments of inertia of the disk about each of the three coordinate axes, assuming constant area density

e. Find the length and the centroid of the semicircular arc in the first quadrant.

f. Find the center of mass and the moments of inertia of the disk if the density is r.

g. Find the area common to the disk sketched and the diskrole="math" localid="1659192769874" $\mathit{r}\mathbf{\le }\mathit{a}$.

For a thin rod of length and uniform densityfind

(a) M,

(c)${\mathbf{l}}_{\mathbf{m}}$about an axis perpendicular to the rod,

(d)labout an axis perpendicular to the rod and passing through one end (see Problem 1).

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{1}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{2}}^{\mathbf{1}}\mathbf{8}\mathbf{xy}\mathbf{}\mathbf{dxdy}\mathbf{.}$