1.sketch the graph of f(x)= x + 2x - 1between x= -1 and x=2 . 1 &f >/0 yes, A=3. otherwise, no 2 2. Evaluate S (x + x + x - 1)dx using the limit definition of the definite integral 3. 0 3 + 7 . ( xy ) 2 A = TR - T (y -o ) = TT ( Vx ) = TX ( x , 0 9 " x V- lux dx - ..- Do at home . 4. Find the volume of the solid using cross section method S A(y) dy. . Example 5: Find the volume of the solid generated when the area bounded by y = a2, the y-axis, and the line y = 4 is revolved about the y-axis using the cylindrical shell method. AV - 27x-(4-y).Ax = 2xx(4-x2) Ax = 27(4x-23) 4x So, the volume is then A= arch - 27x (4-y ) A = 27 ( 4x-X ) * 2 v = [ 27 (4x-23) da = 27 (4x-23) dx. = V = 21 ( 2.(2 - -24 )|- = 87. Note : Tay at home using 6 7 y = 2 2 cross - section method: (2, 4) h= 4 - 4 V = [ Aly ) dy 2 (x, y) * 2 TR -2 2 A R- ?0 Example 6: The area bounded by :1: = 3/2 and the line :1: = 4 is revolved about the line a: = 2. Find the volume of the solid menerated by this area using the cylindrical shell method. X f(X) AV~ 2w)(-(2y)-__=Ag; 27r(:c+2)(2\\/_)A$= 47r(x3/2+2331/2)A$ X--2( ) heigth= 2y So, the volume Is then find y 6. . Example 7: The area bounded by y = x2 and a = y2 is revolved about the line y = -1. Find the volume of the solid generated using the cylindrical shell method. Y- (-1) AV ~ 27 (y+ 1) . (Vy=y'). Ay = 2n(3/2-23 +yl/2-y?) Ay So, the volume is then V = 27(23/2 -y3 ty! /2-y?) dy = 27 (13/2 -y3 ty!/2-y?) dy 1 4 2 29TT - V = 27 5/2 y 3/2 30 y =22 2 Note : croes. Section : x = Uz Ay 01 1 V = ACY ) dx 2 y =-1 -2 LOOK AT THE red loine0 Example 4: Find the area bounded by the curve y = f($)= 3x 32,6 and the x axis. xfbax>=o 0 Solution: Pr: JEAM $841) 5x AX: \"5334; v0 '5 7 7Exercises: 2. 2 2. 4 a and b only), 2.10 on pages 57 59