Find the volume of the solid about y-axis, i.e. 1/(x^2+3x+2), bounds x=0, x=1

Question

find intersection points: 1/(x^2+3x+2)=1, so x=(sqrt(5)-3)/2, x= -(sqrt(5)+3)/2

thus we have:

V(volume)= Pi*integral((1)^2-(1/(x^2+3x+2)^2) from x= -(sqrt(5)+3)/2, x= (sqrt(5)-3)/2

integral((1)^2-(1/(x^2+3x+2)^2)= (2x+3)/(x+1)(x+2) + x + 2log((x+1)/(x+2))

Could somebody tell me, Am I on the right path?

Answer 1

Hint: I strongly suggest using the method of Cylindrical Shells. The volume will be $$\int_0^1 2\pi \frac{x}{x^2+3x+2}\,dx.$$ For the integration, use partial fractions.

Remark: There are errors in the attempt to do it by slicing. It is possible to do it that way, but it will be quite hard. The integration will need to be with respect to $y$.