8.2 Area of a Surface of

8.2 Area of a Surface of - 8.2 Area of a Surface of...

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8.2 Area of a Surface of Revolution Definition: If the graph of a continuous function is revolved about a line, the resulting surface is called a surface of revolution Axis of Revolution L 1 r 2 r ) ( 2 1 radius average segment line of length where 2 2 1 r r r L rL S + = = = = π Surface Area of a Frustum of a Cone:
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Surface Area: = yds S π 2 If the curve is rotated about the x -axis = xds S π 2 If the curve is rotated about the y -axis Note: ) ( when 1 2 x f y dx dx dy ds = + = Use or ) ( when 1 2 y g x dy dy dx ds = + =
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Examples: Find the area of the surface obtained by rotating the curve about the x -axis: 1) 8 0 , 4 4 2 + = x x y Solutions: ( 29 ( 29 ( 29 ( 29 ( 29 2 10 5 3 16 8 40 3 4 3 2 2 4 4 4 2 2 0 when : Limit Lower 6 8 when : Limit Upper 4 4 4 1 1 2 2 4 1 1 Use 4 4 4 4 Now, 2 have We 2 / 3 2 / 3 6 2 2 / 3 2 2 / 1 6 2 2 6 2 2 2 2 2 2 2 2 - = - = + = + = + = = = = = + = + = + = = + = - = + = = π π π π π π y dy y y dy y
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