Chapter 5: Multiple Integrals

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

Find the volume between the surfaces $\mathit{z}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{12}$and $\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{8}$, and over the triangle with vertices (0,0), (1,0) and (1,2).

Find the mass of the solid in Problem 42 if the density is proportional to y.

Find the mass of the solid in Problem 43 if the density is proportional to x.

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

Find the volume in the first octant bounded by the coordinate planes and the plane $\mathit{x}\mathbf{\text{+ 2}}\mathit{y}\mathbf{\text{+}}\mathit{z}\mathbf{\text{= 4}}$.

Find the volume in the first octant bounded by the cone${\mathit{z}}^{\mathbf{2}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}$ and the plane $\mathit{x}\mathbf{=}\mathbf{4}$ .

Find the volume in the first octant bounded by the paraboloid $\mathit{z}\mathbf{=}\mathbf{1}\mathbf{-}{\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}$, the plane $\mathit{x}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{1}$, and all three coordinate planes.

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer.

${\mathbf{\int }}_{\mathbf{x}\mathbf{-}\mathbf{0}}^{\mathbf{4}}{\mathbf{\int }}_{\mathbf{y}\mathbf{-}\mathbf{0}}^{\mathbf{x}\mathbf{/}\mathbf{2}}\mathbf{y}\mathbf{-}\mathbf{dy}\mathbf{-}\mathbf{dx}$

$\mathbf{\int }\sqrt{\mathbf{1}\mathbf{+}\mathbf{"}\mathbf{"}\mathbf{a}{\mathbf{"}}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{x}}{\mathbf{2}}\sqrt{\mathbf{1}\mathbf{+}\mathbf{"}\mathbf{"}\mathbf{a}{\mathbf{"}}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}\mathbf{a}}\mathit{s}\mathit{i}\mathit{n}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$