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barcode print in asp net APPLICATIONS OF INTEGRATION II: VOLUME in .NET framework
APPLICATIONS OF INTEGRATION II: VOLUME QR Code JIS X 0510 Recognizer In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. QR Code 2d Barcode Generation In VS .NET Using Barcode generation for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. the cylinders obtained by rotating the rectangles with the same height f (xi ) and with bases [0, xi 1 ] and [0, xi ]. Hence, it has volume 2 2 2 2 xi f (xi ) xi 1 f (xi ) = f (xi )(xi xi 1 ) = f (xi )(xi + xi 1 )(xi xi 1 ) = f (xi )(2xi )( x) = 2 xi f (xi ) x Read Denso QR Bar Code In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Bar Code Drawer In .NET Framework Using Barcode maker for .NET framework Control to generate, create bar code image in .NET applications. Thus, the total V is approximated by
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Fig. 3315 33.7 (Solids of Revolution about Lines Parallel to a Coordinate Axis) If a region is revolved about a line parallel to a coordinate axis, we translate the line (and the region along with it) so that it goes over into the coordinate axis. The functions de ning the boundary of the region have to be recalculated. The volume obtained by revolving the new region around the new line is equal to the desired volume. (a) Consider the region R bounded above by the parabola y = x 2 , below by the xaxis, and lying between x = 0 and x = 1 [see Fig. 3316(a)]. Find the volume obtained by revolving R around the horizontal line y = 1. (b) Find the volume obtained by revolving the region R of part (a) about the vertical line x = 2. (a) Move R vertically upward by one unit to form a new region R . The line y = 1 moves up to become the xaxis. R is bounded above by y = x 2 + 1 and below by the line y = 1. The volume we want is obtained by revolving R APPLICATIONS OF INTEGRATION II: VOLUME
[CHAP. 33
Fig. 3316 about the xaxis. The washer formula applies, V = =
((x 2 + 1)2 12 ) dx =
(x 4 + 2x 2 ) dx 13 15
2 x5 + x3 5 3
1 2 + 5 3 (b) Move R two units to the right to form a new region R # [see Fig. 3316(b)]. The line x = 2 moves over to become the yaxis. R # is bounded above by y = (x 2)2 and below by the xaxis and lies between x = 2 and x = 3. The volume we want is obtained by revolving R # about the yaxis. The cylindrical shell formula applies, V = 2 = 2 x(x 2)2 dx = 2
(x 3 4x 2 + 4x) dx = 2
1 4 4 3 x x + 2x 2 4 3 = 2 11 12
1 4 4 3 (3) (3) + 2(3)2 4 3 1 4 4 3 (2) (2) + 2(2)2 4 3 11 = 6 Supplementary Problems
Strategy: In calculating the volume of a solid of revolution we usually apply either the disk formula (or the washer formula) or the cylindrical shells formula (or the difference of cylindrical shells formula). To decide which formula to use: (1) Decide along which axis you are going to integrate. This depends on the shape and position of the region R that is revolved. (2) (i) Use the disk formula (or the washer formula) if the region R is revolved perpendicular to the axis of intergration. (ii) Use the cylindrical shells formula (or the difference of cylindrical shells formula) if the region R is revolved parallel to the axis of integration. 33.8 Find the volume of the solid generated by revolving the given region about the given axis. (a) The region above the curve y = x 3 , under the line y = 1, and between x = 0 and x = 1; about the xaxis. (b) The region of part (a); about the yaxis. (c) The region below the line y = 2x, above the xaxis, and between x = 0 and x = 1; about the yaxis. (d) The region between the parabolas y = x 2 and x = y2 ; about either the xaxis or the yaxis. CHAP. 33]

