volume of solid rotated around axis $y=-1$ region bounded by $y=x^3$, $x=1$, $y=-1$ Using a sketch show how to approx. volume by a Riemann sum, and find volume..
We learnt to find the volume by finding the area of cross section and taking the integral of that area
I tried $\int_{-1}^1 \pi (x^3+1)^2dy$ but its wrong
The answer according to wolfram alpha is $24\left(\frac\pi5\right)$, but they used different method I think (not the "Riemann sum" way)

 A: First of all, to me your formula to get the volume is correct. So, what is wrong here? Wolfram Alpha uses the method of Shell integration (see http://en.wikipedia.org/wiki/Shell_integration and http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-Volums-CylinShells_Stu%20.pdf).
The idea, instead of integrating the area of circles contained in the volume of revolution, is to integrate the area of cylinders in the volume of revolution. The formula used by Wolfram Alpha is
$$V=2\pi\int_{-1}^1 (1-y)(1-y^3)dx.$$ Here, $1-y$ is the height of the cilinder and $1-y^3$ is the radius (the cylider has as axis the line $y=-1).$ In my opinion this formula is wrong. If I am not wrong the radius of the cylinder should read $1-\sqrt[3]{y}$ (that is the distance from the point $(x,y)=(x,x^3)=(\sqrt[3]{y},y)$ to the point $(1,y))$ and the height of the cylinder should read $1+y$ (that is, the distance from point $(\sqrt[3]{y},-1)$ to the point $(\sqrt[3]{y},y)).$ That is, in my opinion, the correct formula is 
$$V=2\pi\int_{-1}^1 (y+1)(1-\sqrt[3]{y})dx.$$ After a change of variable this can be written as 
$$V=2\pi\int_{0}^2 y(1-\sqrt[3]{y-1})dx.$$ 
I am not an expert in Mathematica, so I cannot say the reason for this wrong answer. Even, if we ask Wolfram Alpha to get the volume of a cone of radius $1$ and height $1$ gives as answer $2\pi/3:$ http://www.wolframalpha.com/input/?i=revolve+region+between+y%3Dx%2C+y%3D0%2C+x%3D1+about+line+y%3D0 Again, this is not the expected result. 
Note that this formula gives the same result that you should obtain by using your formula.
