Chapter 5: Multiple Integrals

Question: A partially silvered mirror covers the square area with vertices at $\mathbf{(}\mathbf{\pm }\mathbf{1}\mathbf{,}\mathbf{\pm }\mathbf{1}\mathbf{)}$ . The fraction of incident light which it reflects at (x, y) is$\frac{{\left(x-y\right)}^{\mathbf{2}}}{\mathbf{4}}$. Assuming a uniform intensity of incident light, find the fraction reflected.

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{1}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{x}}^{\mathbf{2}\mathbf{x}}{\mathbf{\int }}_{\mathbf{z}\mathbf{=}\mathbf{0}}^{\mathbf{y}\mathbf{-}\mathbf{x}}\mathit{d}\mathit{z}\mathit{d}\mathit{y}\mathit{d}\mathit{x}$

${\mathbf{\int }}_{\mathbf{z}\mathbf{=}\mathbf{0}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{z}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{8}\mathbf{x}}^{\mathbf{z}}\mathbf{dydxdz}$

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{2}}^{\mathbf{3}}{\mathbf{\int }}_{\mathbf{z}\mathbf{=}\mathbf{1}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{y}\mathbf{+}\mathbf{z}}^{\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{z}}\mathbf{6}\mathbf{ydxdzdy}$

Find the moment of inertia of a circular disk (uniform density) about an axis through its centre and perpendicular to the plane of the disk.

A thin rod 10 ft long has a density which varies uniformly from 4 to 24 lb/ft. Find

(a) M,

(b)$\overline{\mathbf{x}}$,

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

(d)labout an axis perpendicular to the rod and passing through the heavy end.

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{2}\mathbf{y}}^{\mathbf{4}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$

Find the area of the paraboloid${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{z}$inside the cylinderrole="math" localid="1659151613290" ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{9}$

Evaluate the triple integrals ${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{1}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{z}\mathbf{=}\mathbf{x}}^{\mathbf{2}\mathbf{x}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{1}\mathbf{/}\mathbf{z}}\mathit{z}\mathit{d}\mathit{y}\mathit{d}\mathit{z}\mathit{d}\mathit{x}$.

Find the volume between the planes z = 2x + 3y +6 and z = 2x + 7y + 8, and over the triangle with vertices, (0,0) (3,0) and (2,1).