Chapter 5: Multiple Integrals

Repeat Problem 23a for the paraboloid$\frac{\mathbf{z}}{\mathbf{c}}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{2}}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{2}}}{{\mathbf{b}}^{\mathbf{2}}}\mathbf{}\mathbf{}\mathbf{0}\mathbf{<}\mathbf{z}\mathbf{<}\mathbf{c}\mathbf{.}$

Under the surface z = 1 /(y+2) , and over the area bounded by and y=x ${\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{2}$.

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 inertia about the -axis of a lamina in the shape of the plane area under the curve; of a wire bent along the arc of the curve.

The volume inside a sphere of radius ris$\mathbf{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\tau \tau r}}^{\mathbf{3}}$. Then$\mathbf{dV}\mathbf{=}\mathbf{4}{\mathbf{\tau \tau r}}^{\mathbf{2}}\mathbf{dr}\mathbf{=}\mathbf{Adr}$whereis the area of the sphere. What is the geometrical meaning of the fact that the derivative of the volume is the area? Could you use this fact to find the volume formula given the area formula?

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 inertia about the x axis of a lamina in the shape of the plane area under the curve;

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{3}\mathbf{-}\mathbf{3}\mathbf{x}}\mathbf{dydx}$

Use the parallel axis theorem (Problem 3.1)

(a) and Example 3, to find the moment of inertia of a solid ball about a line tangent to it;

(b) and Problem 3b to find the moment of inertia of a solid cylinder about a ruling

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{y}\mathbf{/}\mathbf{2}}^{\mathbf{1}}\left(x+y\right)\mathbf{dxdy}$

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

The moments of inertia of a wire bent along the arc of the curve.

Make the change of variables to evaluate the integral${\mathit{\int }}_{\mathit{0}}^{\mathit{1}}\mathit{d}\mathit{x}{\mathit{\int }}_{\mathit{0}}^{\mathit{x}}\frac{\mathit{(}\mathit{x}\mathit{+}\mathit{y}\mathit{)}{\mathit{e}}^{\mathit{x}\mathit{+}\mathit{y}}}{{\mathit{x}}^{\mathit{2}}}\mathit{d}\mathit{y}$