Schaum's Outline of Theory and Problems of Beginning Calculus

264 APPLICATIONS OF INTEGRATION 11: VOLUME [CHAP. 33(noncircular) disk, of thickness Ax and base area A(xF) (see Fig. 33-15). This disk has volume A(x:)Thus,Ax.Fig. 33-1533.7 (Solids of Revolution about Lines Parallel to a Coordinate Axis) If a region is revolved about aline parallel to a coordinate axis, we translate the line (and the region along with it) so that itgoes over into the coordinate axis. The functions defining the boundary of the region have to berecalculated. The volume obtained by revolving the new region around the new line is equal tothe desired volume.(a) Consider the region 9 bounded above by the parabola y = x2, below by the x-axis, andlying between x = 0 and x = 1 [see Fig. 33-16(a)]. Find the volume obtained by revolving Waround the horizontal line y = - 1.(b) Find the volume obtained by revolving the region W of part (a) about the vertical linex= -2.(a)Move W vertically upward by one unit to form a new region W*. The line y = - 1 moves up to becomethe x-axis. W* is bounded above by y = x' + 1 and below by the line y = 1. The volume we want isobtained by revolving W* about the x-axis. The-washer formula applies,V = III'((x' + I)' - 1') dx = II(b) Move W two units to the right to form a new region 92' [see Fig. 33-16(b)]. The line x = -2 movesover to become the y-axis. Wx is bounded above by y = (x- 2)' and below by the x-axis and liesYY1IX- a1 2 3 xFig. 33-16