91Ó°ÊÓ

For Exercises \(1-4,\) calculate \(\int_{C} f(x, y) d s\) for the given function \(f(x, y)\) and curve \(C\). \(f(x, y)=x+y^{2} ; \quad C:\) path from (2,0) counterclockwise along the circle \(x^{2}+y^{2}=4\) to the point (-2,0) and then back to (2,0) along the \(x\) -axis

Show that the flux of any constant vector field through any closed surface is zero.

Evaluate the surface integral \(\iint \mathbf{f} \cdot d \boldsymbol{\sigma},\) where \(\mathbf{f}(x, y, z)=x^{2} \mathbf{i}+x y \mathbf{j}+z \mathbf{k}\) and \(\Sigma\) is the part of the plane \(6 x+3 y+2 z=6\) with \(x \geq 0, y \geq 0,\) and \(z \geq 0,\) with the outward unit normal \(n\) pointing in the positive \(z\) direction.

Use a surface integral to show that the surface area of a sphere of radius \(r\) is \(4 \pi r^{2}\), Hint: Use spherical coordinates to parametrize the sphere.)

The ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) can be parametrized using ellipsoidal coordinates $$ x=a \sin \phi \cos \theta, y=b \sin \phi \sin \theta, z=c \cos \phi, \text { for } 0 \leq \theta \leq 2 \pi \text { and } 0 \leq \phi \leq \pi . $$ Show that the surface area \(S\) of the ellipsoid is $$ S=\int_{0}^{\pi} \int_{0}^{2 \pi} \sin \phi \sqrt{a^{2} b^{2} \cos ^{2} \phi+c^{2}\left(a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta\right) \sin ^{2} \phi} d \theta d \phi $$

For Exercises \(14-15\), verify Stokes' Theorem for the given vector field \(\mathbf{f}(x, y, z)\) and surface \(\Sigma\). $$ \mathbf{f}(x, y, z)=2 y \mathbf{i}-x \mathbf{j}+z \mathbf{k} ; \quad \Sigma: x^{2}+y^{2}+z^{2}=1, z \geq 0 $$