5.10 Parameterizations of Surfaces and Surface Area (continued)

The Surface Area of Parametrically Defined Surfaces

Work through this exercise and explain your reasoning step by step.

Calculate $\vr_s, \vr_t, \vecmag{\vr_s \times \vr_t}$ based on the parameterization given above.

Use the calculations from the previous part to set up an iterated integral to determine the surface area of this cylinder.

Evaluate your iterated integral from the previous part.

One way to think about the surface area of a cylinder is to cut the cylinder horizontally and find the perimeter of the resulting cross sectional circle, then multiply by the height. Calculate the surface area of the given cylinder using this alternate approach, and compare your result the value from the previous part.

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

.

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 $D = [-2,2] \times [0,3]$.

Observe that the surface of the solid describe the previous part is half of a circular cylinder. Use the standard formula for the surface area of a cylinder to calculate the surface area in a different way, and compare your result from above.

Consider the ellipsoid given by the equation

In Activity 5.10.1, 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 $0 \leq t \leq 2\pi$. - 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 $t$. - Use the result of part (a) to find a parameterization of the ellipse in terms of $x$, $y$, and $z$ as functions of $s$ and $t$. Check your parametrization by substituting $x$, $y$, and $z$ into the equation of the ellipsoid. Then check your work by plotting the surface defined by your parameterization.

In this exercise, we explore how to use a parametrization and iterated integral to determine the surface area of a sphere. - 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 5.10.1). - Then, evaluate the integral to calculate the surface area of this portion of the sphere. - By what constant must you multiply the value determined in (b) in order to find the total surface area of the entire sphere. - Finally, compare your result to the standard formula for the surface area of sphere.

Consider the plane generated by $z = f(x,y) = 24 - 2x - 3y$ over the region $D = [0,2]\times[0,3]$. - Sketch a picture of the overall solid generated by the plane over the given domain. - Determine a parameterization $\vr(s,t)$ for the plane over the domain $D$. - Use Equation to determine the surface area generated by $f$ over the domain $D$. - 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 $\vv$ such that $\vu$ and $\vv$ together span the parallelogram that represents the surface defined by $f$ over $D$, and hence compute $| \vu \times \vv |$. What do you observe about the value you find?

A cone with base radius $a$ and height $h$ can be realized as the surface defined by $z = \frac{h}{a} \sqrt{x^2+y^2}$, where $a$ and $h$ are positive. - 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 5.10.2.) - Set up an iterated integral to determine the surface area of this cone. - Evaluate the iterated integral to find a formula for the lateral surface area of a cone of height $h$ and base $a$.