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6.1.47-eg-1 - or r x = 1-2 = 3 The area of a slice is A x =...

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Example Find the volume of the solid generated by rotating the region bounded by y = x 2 , y = 1, and x = 4 about the line y = - 2. See figure below (not drawn to scale!). (a) Find the area of a slice. Since the region does not border the axis of rotation, each slice will be a “washer” (annulus) with vertical outer and inner radii perpendicular to the axis of rotation ( y = - 2). For fixed x , the (outer) radius R is the distance from the the top boundary of the region to the axis of rotation, or R ( x ) = x 2 - ( - 2) = x 2 + 2, whereas the (inner) radius r is given by the distance from the bottom boundary of the region to the axis of rotation,
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Unformatted text preview: or r ( x ) = 1-(-2) = 3. The area of a slice is A ( x ) = πR ( x ) 2-πr ( x ) 2 = π ( x 2 + 2) 2-π (3) 2 . (b) Find the limits of integration. The washers run from the x-coordinate of the intersection of y = x 2 with y = 1 in the first quadrant, so from x = 1, to x = 4. (c) Find the volume of the region. The volume V is given by V = π Z 4 1 ( x 2 + 2) 2-9 dx = π x 5 5-4 x 3 3-5 x ! ± ± ± ± 4 1 = 1368 π 5 ....
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