Definite integral (Volume)

Consider a small elemental cross-sectional area, dA = pi y^2 = pi (x^2 - 1)^2

When it is rotated around the y axis, the volume element, dV = dxdA

= pix (x^2 - 1)^2 dx

= pi x(x^4 - 2x^2 + 1) dx

= pi(x^5 - 2x^3 + x) dx

So, integrate the expression from x = 1 to x = 2

Volume = int (1 -> 2) pi(x^5 - 2x^3 + x) dx

= pi [x^6/6 - x^4/2 + x^2/2] (1 -> 2)

= pi {[(2)^6/6 - (2)^4/2 + (2)^2/2] - [(1)^6/6 - (1)^4/2 + (1)^2/2]}

= 9pi / 2

in from x=1, to x=2 (pi (x^2-1)^2 ) dx=9pi/2

Sincerely, Simon YAU
http://hk-math.com