Homework Soursefind the volume of the region bounded by y2x2 yxx0 and the y
find the volume of the region bounded by y=2-x^2, y=x(x>0) and the y axis revolved around y=-1
Solution
Shell method: V = 2p ? r h dr {a,b} .......... limits in {} radius is (1+y) to rotate around y=-1 = 2p ?(1+y)(2) dx {0,1/3} + 2p ?(1+y)(1/y - 1) dx {1/3,1} = 4p [y + y²/2] {0,1/3} + 2p [ ln(y) - y²/2 ] {1/3,1} = 14p/9 + 2p [ ln(3) - 4/9 ] = 2p [ ln(3) + 1/3 ] ˜ 8.997 units³ Disc method: V = p ? f(x)² dx {a,b} rotate around y=-1 extends the inner and outer radii by 1 = p ? (1/x + 1)² - (1)² dx {1,3} .............. note: p [(Ro)² - (Ri)²] = p ? 1/x² + 2/x dx {1,3} = p [ -1/x + 2 ln(x) ] {1,3} = p [ -1/3 + 2 ln(3) + 1 ] = 2p [ ln(3) + 1/3 ] Answer: ˜ 8.997 units³
