# Math Help - volume about the y axis

1. ## volume about the y axis

Hi, the questions is to find the volume of the solid formed when the curve y=x^3 rotated about the y axis between y=0 and y=1. I am having trouble integrating once x^2 has been made the subject of the equation.
y=x^3
y/x=x^3/x
therefore
x^2=y/x

Is this right? If yes, how do you integrate an expression with both an x and y value? I haven't come across this before. Cheers

2. We'll use the disc method. Since we're rotating around the $y$-axis, we're going to be integrating with respect to $y$

So, expressing the equation in terms of y: $y = x^3 \ \Leftrightarrow \ x = \sqrt[3]{y}$

Each disk has a radius of $x$. So the area of each disk in terms of y, i.e. $A(y)$, is $\pi x^2 = \pi \left(\sqrt[3]{y}\right)^2 = \pi y^{\frac{2}{3}}$

So its volume is given by: $V = \int_{y = 0}^{y = 1} A(y) dy = \int_0^1 \pi y^{\frac{2}{3}} dy$