I was trying to derive the formula for obtaining the surface area of a sphere, but I kept making mistakes whereby I chopped the surface in dx or dy. When you calculate the arc length of a curve or the surface area of a volume, you have to chop it in lengths of the slope, not the width or the height.
To compute the length of an arc of a curve, we need to define the differential of arc length. If we zoom in on a small section of a curve, we find that the length of a curve s is the length of the slope of a right triangle whose base is dx and whose height is dy.
Therefore,
Surfaces of Revolution
Using the differential of arc length, we can find the area of the surface generated by rotating a curve C about the x-axis or y-axis. When a curve of function y=f(x) is rotated about the x-axis, the surface area of the generated solid can be found with the following formula:
When a curve of function y=f(x) is rotated about the y-axis, the surface area of the generated solid can be found with the following formula:
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