Maths - Application of Integration http://upload.lsforum.net/users/public/o249271797398578_nw40.jpg?

Given y1 = 1 + x^2 & y2 = 5x - 5

Find volume between the both revoluting y-axis

Solution:

The common point is at (2, 5).

Volume = Pure y1 + Mixed y1 & y2 = V1 + V2

V1 = ∫(0~1)2πx*y1*dx

= ∫2πx(x^2 + 1)dx

= ∫2π(x^3 + x)dx

= 2π(x^4/4 + x^2/2)

= 2π(1/4 + 1/2)

= 3π/2


V2 = ∫2πx(y1 - y2)dx

= ∫(1~2) 2πx(x^2 + 1 - 5x + 5)dx

= ∫2π(x^3 - 5x^2 + 6x)dx

= 2π(x^4/4 - 5x^3/3 + 3x^2)

= 2π(15/4 - 5*7/3 + 3*3)

= 2π/12 * (15*3 - 35*4 + 9*12)

= π/6 * (45 + 108 - 140)

= 13π/6


V = V1 + V2

= 3π/2 + 13π/6

= 22π/6

= 11π/3

= Answer