91Ó°ÊÓ

(a) Write a triple integral in cylindrical coordinates for the volume of the part of a ball between two parallel planes which intersect the ball.

(b) Evaluate the integral in (a). Warning hint: Do the r and$\mathit{θ}$integrals first.

(c) Find the centroid of this volume.

Prove the following two theorems of Pappus: An arc in the (x,y)plane,$\mathit{y}\mathbf{â‰\yen }\mathbf{0}$, is revolved about the x axis. The surface area generated is equal to the length of the arc times the circumference of the circle traced by the centroid of the arc.

$\mathbf{∫}\mathbf{∫}\mathbf{2}\mathit{x}\mathit{y}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$over the triangle with vertices role="math" localid="1658828431182" $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{(}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{0}\mathbf{)}$.

$\mathbf{âˆ\neg }{\mathit{x}}^{\mathbf{2}}{\mathit{e}}^{{\mathbf{x}}^{\mathbf{2}}\mathbf{y}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$over the area bounded by$\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{-}\mathbf{x}}\mathbf{,}\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{-}\mathbf{2}}$and$\mathit{x}\mathbf{=}\mathit{I}\mathit{n}\mathbf{4}$

Express the integral$\mathit{I}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{1}}\mathit{d}\mathit{x}{\mathbf{∫}}_{\mathbf{0}}^{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}\mathbf{}{\mathit{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}}\mathit{d}\mathit{y}$as an integral in polar coordinates $\left(r,\mathrm{θ}\right)$ and so evaluate it.

Prove the following two theorems of Pappus: Use Problems 12 and 13 to find the volume and surface area of a torus (doughnut).

Express the integral $\mathit{I}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{1}}\mathit{d}\mathit{x}{\mathbf{∫}}_{\mathbf{0}}^{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}\mathbf{}{\mathit{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}}\mathit{d}\mathit{y}$as an integral in polar coordinates $\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{θ}\mathbf{)}$ and so evaluate it.

Use Problems 12 and 13 to find the centroids of a semi-circular area and of a semi-circular arc. Hint: Assume the formulas $\mathit{A}\mathbf{=}\mathbf{4}\mathit{Ï€}{\mathit{r}}^{\mathbf{2}}$, $\mathit{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathit{Ï€}{\mathit{r}}^{\mathbf{3}}$ for a sphere.

$\mathbf{âˆ\neg }\mathit{d}\mathit{x}\mathbf{}\mathit{d}\mathit{y}$over the area bounded by$\mathit{y}\mathbf{=}\mathbf{Inx}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{e}\mathbf{+}\mathbf{1}\mathbf{-}\mathbf{x}$ and the x axis.

Find the centroid of the first quadrant part of the arc ${\mathbf{x}}^{\raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}\mathbf{+}{\mathbf{y}}^{\raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}\mathbf{=}{\mathbf{a}}^{\raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}$ . Hint: Let$\mathbf{x}\mathbf{=}\mathbf{a}\mathbf{}\mathbf{cos}\mathbf{3}\mathbf{}\mathbf{θ}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{a}\mathbf{}\mathbf{sin}\mathbf{3}\mathbf{}\mathbf{θ}$ .