Find the area of the surface obtained by rotating the curve
y=1+5x^2
from x=0 to x=1 about the y-axis
I know how to do it about the x-axis but how does the formula change if your rotating about the y axis
The formula you need is
$\displaystyle S=\int_{a}^{b}2 \pi x \sqrt{1+\left( \frac{dx}{dy}\right)^2}dy$
First to find the limits of integration we plug in x=0 and x=1 to get
y=1 and y=6
Now we solve the equaiton for x to get
$\displaystyle y=1+5x^2 \iff x=\sqrt{\frac{y-1}{5}}=\frac{1}{\sqrt{5}}(\sqrt{y-1})$
Now we take the derivative with respect to y to get
$\displaystyle \frac{dx}{dy}=\frac{1}{2\sqrt{5}}\frac{1}{\sqrt{y-1}}$
From here just plug into the formula and integrate.
Best Wishes