91Ó°ÊÓ

Find the integral of $f(x,y,z)=z^3+z\left(x^2+2^y\right)$on the portion of the unit sphere that lies in the first octant, above the rectangle $\left[0,\frac{1}{2}\right]×\left[0,\frac{1}{3}\right]$ in the XY-plane.

Evaluate the integrals in Exercises 43–48.

Find the integral of $$f(x, y,z) = z^{3} + z(x^{2} + 2^{y})$$ on the portion of the unit sphere that lies in the first octant, above the rectangle $$[0,\frac{1}{2}] \times [0,\frac{1}{3}]$$ in the xy-plane.

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y,z)=\mathbf{i}+2\mathbf{j}−3\mathbf{k}$

Evaluate the integrals in Exercises 43–46 directly or using Green’s Theorem.

46. Suppose that an electric field is given by $\mathbf{E}=⟨2y,2xy,yzâŸ\copyright$. Use the Divergence Theorem to compute the flux ${âˆ\neg }_S\mathbf{E}d\mathbf{A}$ of the field through the surface of the unit cube $[0,1]×[0,1]×[0,1]$.

(a) Use Stokes' Theorem and the Fundamental Theorem of Line Integrals to show that
${âˆ\neg }_{S}curl\mathbf{F}(x,y,z)·\mathbf{n}dS=0$

for any conservative vector field $\mathbf{F}$.

(b) Without using Stokes' Theorem, show that

${âˆ\neg }_{S}curl\mathbf{F}(x,y,z)·\mathbf{n}dS=0$

for any conservative vector field$\mathbf{F}$.

Find ${∫}_S 1dS$, where S is the portion of the surface with equation $x=e^{yz}−e^{−yz}$that lies on the positive side of the circle of radius 3 and centered at the origin in the yz-plane.

Find the work done by the vector field

$\mathbf{F}(x,y)=\left(\cos \left(x^2\right)+4x{y}^2\right)\mathbf{i}+\left(2^y-4x^{2}y\right)\mathbf{j}$

in moving an object around the unit circle, starting and ending at $(1,0)$.

Find

$\begin{array}{r}{∫}_S \mathbf{F}(x,y,z)â‹\dots \mathbf{n}dS\text{if}\\ \mathbf{F}(x,y,z)=2xz\mathbf{i}+2yz\mathbf{j}−18\mathbf{k}\end{array}$

and S is the portion of the hyperboloid $x^2+y^2-9=z^2$that lies between the planes

z = −4 and z = 0, with n pointing outwards.

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{G}(x,y,z)=(z−y)\mathbf{i}−xy\mathbf{j}+(xz+y)\mathbf{k}$