Volumes of Revolution around the y-axis
Calculating around the y-axis
- This is the same as calculating around the y axis except you rearrange the equation to get \(x\) alone in terms of \(y\).
- You then either use \(x^2\) in your integral, or use \(x\) and then square it inside the integral.
\(y=\sqrt{ x-1 }\)
- \(y=\sqrt{ x-1 }\)
- \(x=y^2+1\)
- \(x^2=(y^2+1)^2\)
- \(x^2=y^4+2y^2+1\)
- \(V=\pi \int ^3_{1}y^4+2y^2+1 \, dy\)
- \(V=\pi\left[ \frac{1}{5}y^5+\frac{2}{3}y^3+y \right]^3_{1}\)
- \(V=\frac{1016\pi}{15}\)
\(y=\sqrt[3{2x+1}\)
- \(y=\sqrt[3]{2x+1}\)
- \(y^3=2x+1\)
- \(2x=y^3-1\)
- \(x=\frac{y^3}{2}-\frac{1}{2}\)
- \(x^2=\left( \frac{y^3}{2}-\frac{1}{2} \right)^2\)
- \(x^2=\left( \frac{y^3}{4}-\frac{1}{4} \right)\)
- \(\pi \times \int ^4_{2} \left( \frac{1}{2}y^3-\frac{1}{2} \right)^2\, dy=\frac{7715\pi}{14}\)