Parameterizations of Surfaces and Surface Area

1. Let $D$ be a region in the domain of $f$. Using equation, show that the area of the surface defined by the graph of $f$ over $D$ is .

2. Use the formula developed in the previous part to calculate the area of the surface defined by $f(x,y) = \sqrt{4-x^2}$ over the rectangle $D = [-2,2] \times [0,3]$.

3. Observe that the surface of the solid describe the previous part is half of a circular cyl…

In Activity, we found that a parameterization of the sphere $S$ of radius $R$ centered at the origin is for $-\frac{\pi}{2} \leq s \leq \frac{\pi}{2}$ and $0 \leq t \leq 2\pi$.

1. Let $(x,y,z)$ be a point on the ellipsoid and let $X = \frac{x}{4}$, $Y = \frac{y}{5}$, and $Z = \frac{z}{3}$. Show that $(X,Y,Z)$ lies on the sphere $S$. Hence, find a parameterization of $S$ in terms of $X$, $Y$, and $Z$ as functions of $s$ and $t$.

2. Use the result of part (a) to find a parameterization of the ellipse…

1. Set up an iterated integral whose value is the portion of the surface area of a sphere of radius $R$ that lies in the first octant (see the parameterization you developed in Activity).

2. Then, evaluate the integral to calculate the surface area of this portion of the sphere.

3. By what…

1. Sketch a picture of the overall solid generated by the plane over the given domain.

2. Determine a parameterization $\vr(s,t)$ for the plane over the domain $D$.

3. Use Equation to determine the surface area generated by $f$ over the domain $D$.

4. Observe that the vector $\vu = \langle 2, 0, -4 \rangle$ points from $(0,0,24)$ to $(2,0,20)$ along one side of the surface generated by the plane $f$ over $D$. Find the vector $\vv$…

1. Find a parameterization of the cone described by $z = \frac{h}{a} \sqrt{x^2+y^2}$. (Hint: Compare to the parameterization of a cylinder as seen in Activity.)

2. Set up an iterated integral to determine the surface area of this cone.

3. Evaluate the iterated integral to find a formula for the lateral surface area of a cone of h…