91Ó°ÊÓ

Using \(f=\left\langle 3 x, y^{3},-2 z^{2}\right\rangle\) and the region bounded by \(x^{2}+y^{2}=9, z=0,\) and \(z=5,\) compute both integrals from the Divergence Theorem.

Let \(f=\langle x y,-x y\rangle\) and let \(D\) be given by \(0 \leq x \leq 1,0 \leq y \leq 1 .\) Compute \(\int_{\partial D} f \cdot d r\) and \(\int_{\partial D} \boldsymbol{f} \cdot \boldsymbol{N} d s\).

Compute \(\int_{\partial D} 2 y d x+3 x d y,\) where \(D\) is described by \(0 \leq x \leq 1,0 \leq y \leq 1 .\)

Describe or sketch the surface with the given vector function. (a) \(\boldsymbol{r}(u, v)=\langle 6 v-u, 3+v, 1-4 v\rangle\) (b) \(\boldsymbol{r}(u, v)=\langle 2 \cos u, 5 \cos u, v\rangle\) (c) \(\boldsymbol{r}(s, t)=\left\langle u+v, 2 u+v, u^{2}+v^{2}\right\rangle\) (d) \(\boldsymbol{r}(s, t)=\langle\sin u+\cos u, u, v\rangle\)

Find the center of mass of an object that occupies the upper hemisphere of \(x^{2}+y^{2}+z^{2}=1\) and has density \(x^{2}+y^{2}\).

Compute \(\int_{C} \sin x d s\) along the line segment from (-1,2,1) to \((1,2,5) .\)

Find the area of the portion of \(x+2 y+4 z=10\) in the first octant.

Investigate the vector field \(\langle-x,-y\rangle .\)

Find the center of mass of an object that occupies the surface \(z=x y, 0 \leq x \leq 1,0 \leq y \leq 1\) and has density \(\sqrt{1+x^{2}+y^{2}} .\)

Compute \(\int_{\partial D} x y d x+x y d y,\) where \(D\) is described by \(0 \leq x \leq 1,0 \leq y \leq 1 .\)