# Thread: volume problem - cylindrical shell method

1. ## volume problem - cylindrical shell method

"The region enclosed by the curves $\displaystyle y=x^{3}$ and $\displaystyle y=\sqrt{x}$ is roated about the line $\displaystyle x=1$. Fint the volume of the resulting solid" -James Stewart

answer:$\displaystyle \frac{13\pi}{30}$
what I did:
difference of shells formula is $\displaystyle 2\pi \int x(f(x)-g(x))$
so in this problem $\displaystyle f(x)=\sqrt{x}$ and $\displaystyle g(x)=x^3$. The intersections are $\displaystyle x=0,1$. I get confused because it's not being rotated around the y-axis but $\displaystyle x=1$
I tried $\displaystyle V= 2 \pi \int_0^1 x((1-\sqrt{x})-(1-x^{3})$ but I end up with a negative number.

2. The x term represents the distance between an arbitrary x and the axis of rotation.

If the solid is rotated about the yaxis the x term is just x.

In your example the distance between x and the axis of rotation x = 1 is

1-x. so replace x with 1-x--- you obtain (1-x)*(sqrt(x)-x^3) for the integrand.

A more general form of the shell method is 2pint{D(x)*(f-g)dx}

Where D(x) is as mentioned the distance between an arbitrary x and the axis of rotation