How do you find the volume of the solid generated by revolving this region about the y axis, x= y^2 and x= y+2?

1 Answer
Sep 21, 2015

V=(72pi)/5

Explanation:

x_1=y^2
x_2=y+2

V=pi int_(y_1)^(y_2) (x_2^2-x_1^2)dy

y_1,y_2=?
x_1=x_2
y^2=y+2
y^2-y-2=0
y^2-y-2=y^2-2y+y-2=y(y-2)+y-2=(y-2)(y+1)=0

y_1=-1, y_2=2

V=pi int_(-1)^(2) ((y+2)^2-(y^2)^2)dy

V=pi int_(-1)^(2) (y^2+4y+4-y^4)dy

V=pi(y^3/3+2y^2+4y-y^5/5)|_-1^2

V=pi(8/3+8+8-32/5+1/3-2+4-1/5)

V=(72pi)/5