Surface Area

# Surface Area - Surface Area In this section we are going to...

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Surface Area In this section we are going to look once again at solids of revolution. We first looked at them back in Calculus I when we found the volume of the solid of revolution . In this section we want to find the surface area of this region. So, for the purposes of the derivation of the formula, let’s look at rotating the continuous function in the interval about the x -axis. Below is a sketch of a function and the solid of revolution we get by rotating the function about the x -axis. We can derive a formula for the surface area much as we derived the formula for arc length . We’ll start by dividing the integral into n equal subintervals of width . On each subinterval we will approximate the function with a straight line that agrees with the function at the endpoints of the each interval. Here is a sketch of that for our representative function using . Now, rotate the approximations about the x -axis and we get the following solid.

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The approximation on each interval gives a distinct portion of the solid and to make this clear each portion is colored differently. Each of these portions are called frustums and we know how to find the surface area of frustums. The surface area of a frustum is given by, where, and l is the length of the slant of the frustum.
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