Chapter 5: Multiple Integrals

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{x}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}{\mathbf{y}}^{\mathbf{2}}\mathbf{/}\mathbf{2}}\mathbf{dydx}$

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

The moment of inertia about y the axis of the solid of revolution if the density is $\mathbf{\left|}\mathit{x}\mathit{y}\mathit{z}\right|$.

(a) Revolve the curve $\mathbf{y}\mathbf{=}{\mathbf{x}}^{\mathbf{1}}$, from $\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{to}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{\infty }$, about the x axis to create a surface and a volume. Write integrals for the surface area and the volume. Find the volume, and show that the surface area is infinite. Hint: The surface area integral is not easy to evaluate, but you can easily show that it is greater than ${\mathbf{\int }}_{\mathbf{1}}^{\mathbf{\infty }}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{dx}$which you can evaluate.

(b) The following question is a challenge to your ability to fit together your mathematical calculations and physical facts: In (a) you found a finite volume and an infinite area. Suppose you fill the finite volume with a finite amount of paint and then pour off the excess leaving what sticks to the surface. Apparently, you have painted an infinite area with a finite amount of paint! What is wrong? (Compare Problem 15.31c of Chapter 1.)

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{l}\mathbf{n}\mathbf{16}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{x}\mathbf{/}\mathbf{2}}}^{\mathbf{4}}\frac{\mathbf{d}\mathbf{y}\mathbf{d}\mathbf{x}}{\mathbf{l}\mathbf{n}\mathbf{y}}$

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}{\mathbf{y}}^{\mathbf{2}}}^{\mathbf{1}}\frac{{\mathbf{e}}^{\mathbf{x}}}{\sqrt{\mathbf{x}}}\mathbf{dxdy}$

Use a computer or tables to evaluate the integral in 3.2and verify that the answer is equivalent to the text answer. Hint: See Problem 1.4 and also Chapter 2 , Sections 15 and 17.

A lamina covering the quarter disk ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{\le }\mathbf{4}\mathbf{,}\mathbf{x}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}$ has (area) density . Find the mass of the lamina.

Verify that (3.10) gives the same result as (3.8).

A dielectric lamina with charge density proportional to y covers the area between the parabola $\mathit{y}\mathbf{=}\mathbf{16}\mathbf{·}{\mathit{x}}^{\mathbf{2}}$ and the x axis. Find the total charge.

A triangular lamina is bounded by the coordinate axes and the line $\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{6}$. Find its mass if its density at each point P is proportional to the square of the distance from the origin to P.