Question

3. Find the volume of the solid generated by revolving the region bounded by the ons y x2 +2, y 0, x- 0, x 3 about the indicated lines: a) the x-axis 4 b) the line x

Answer 1

Lets consider a vertical strip EFGH of thickness dx and height y Now rotating it about x - axis will give a cylinder of width dx and radius y. So, dv = pi y^2 dx given y = x^2 + 2 So, dv = pi(x^2 + 2)^2 dx So, v = integral dv = pi Integral^3_x = 0 (x^2 -+ 2)^2 dx = pi Integral^3_0 (x^4 + 4 + 4x^2) dx = pi [x^5/5 + 4x + 4x^3/3]^3_0 = pi [2 43/5 + 12 + 36] = 483 pi/5.Lets consider a horizontal strip of ABCD rotating it about x = 4 gives a follow cylinder of inner radius 4 - x end outer radius = 4, the the width is dy So, dv = pi(4^2 = -(4 - x)^2 dy = pi(4 - (4 -x) (4 + (4 - x)) dy = pi x(8-x) dy.y = x^2 + 4 dy = 32x dx So, dv = 2 pi x^2 (8 - x)dx.