Area of Surface .pdf

Area of Surface .pdf - Section 9.4 Area of a Surface of...

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Section 9.4: Area of a Surface of Revolution Consider a continuous function f on the interval [ a, b ]. Revolving the curve y = f ( x ), a x b about the x - or y -axis produces a surface known as a surface of revolution . A general formula for the area of such a surface is SA = Z 2 rdL, where L denotes the arc length function and r is the distance from the curve to the axis of revolution (the radius). There are two cases to consider. 1. Revolving about the x -axis. (a) If the curve y = f ( x ), a x b is revolved about the x -axis, then the area of the resulting surface is given by SA = 2 Z b a f ( x ) p 1 + [ f 0 ( x )] 2 dx. (b) If the curve x = g ( y ), c y d is revolved about the x -axis, then the area of the resulting surface is given by SA = 2 Z d c y p 1 + [ g 0 ( y )] 2 dy. (c) If the curve defined by x = x ( t ), y = y ( t ), t β is revolved about the x -axis, then the area of the resulting surface is given by SA = 2 Z β y ( t ) s dx dt 2 + dy dt 2 dt. 2. Revolving about the y -axis. (a) If the curve y = f ( x ), a x b is revolved about the y -axis, then the area of
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