Question

Find the volume of the solid generated by revolving the region R bounded by the graphs of the given equations about the y-axis. 17)x= x=0, between y=- 4 and y = 4 17) 18) bounded by the circle x2 + y2 = 16, by the line x = 4, and by the line y = 4 18) Find the volume of the solid generated by revolving the region about the given line. 19) The region in the first quadrant bounded by the line y = 16, by the curve y = 16 - x2, and by 19). the line x = 4, about the line y = 16. and by the 20) 5x 20) The region in the first quadrant bounded by the line y = 5, by the line y =- y-axis, about the line y = 5. 21) Find the volume of the described solid. 21) The solid lies between planes perpendicular to the x-axis at x = 0 and x = 8. The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola y=-2/x to the parabola y=2Vx. 22) 22) The base of the solid is the disk x2 + y2 s 4. The cross sections by planes perpendicular to the y-axis between y=-2 and y= 2 are isosceles right triangles with one leg in the disk. Find the volume of the solid generated by revolving the region bounded by the given curves about the axis. 23) x = 5w/y,x=- 5y, y=1 23) 5x Find the volume of the solid generated by revolving the region about the given line. 24) The region in the first quadrant bounded above by the line y=5, below by the line and on the left by the y-axis, about the line y = 5 24) 25) 25) The region in the second quadrant bounded above by the curve y=4 - x2, below by the x-axis, and on the right by the y-axis, about the line x = 1 26) 26) The region in the first quadrant bounded above by the line y = 3x3, below by x-axis, and on the right by the line x = 1, about the line y=-1

Answer 1

Sol" : 17 a = y² a=0 4 = -4 to y=4 y = -4 any • Volume = Tot du كل y = 4 4 E TT (422 dy 2 -4 + a=f 16 -2 0 2 4 nx = TT 16 yt dy -2 -4 4 IT T.ly 4 16 15 y=-4 - 4 = TAX + (45-1956) - 128T 5 cube-unit Any 18 . SHELL METHOD, ny Comider 5 a green shaded y = 116-02 M a= 4 rotated region about is being A=4 ren -5 0 5 XV So, radius, r(a) = (4x) height. hn) = 2 (16-012 y = - 116725 O . Volume 21 Sreal. hog da 4 - 2T L (4-7). 2116-92 da -4 = 12872 Cabe unit feny

19 (6) SHELL METHOD, -5 y = 4 Consider a rectanguly strip MM parallel to the rotation axis of m M y = 4. (A = 116-y2 x=-116-52 -5 5 So r(y) = (44) hly) 2 1 16 y -5 Volume of region if rotated about y=4 as parallel to n-axis is given I, hly) dy 211 frcy. (4-9). 2116-42 dy 211 -4 = -2 128 TT Cubic unit Angs