{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

6.1.47-eg-1

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

This preview shows page 1. Sign up to view the full content.

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,
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online