Rotation Around the Y-Axis

Question

I have an equation:

$y = -0.0122625x^2 + 120.38736$

and I want to rotate this around the y-axis and find the volume from the range 0 to 99. I have no idea how to do this and would greatly appreciate some help/explanation of how to do this. Thank you

Answer 1

In general, volumes of curves of the form $y=f(x)$ about the $y$-axis can be carried out in two ways.

(1) Using an integral in $y$, (washer method). This will however require you to invert your function into an expression of the form $x=g(y)$.

(2) Using an integral in $x$, (shell method). This does not require you to invert your function. Check out http://en.wikipedia.org/wiki/Shell_integration for an introduction.

Answer 2

Here is a plot of $y=a x^2 + b$ (red) and the height constraints $y=0$ and $y=99$ (green).

Integrating discs of radius $r = r(y)$ $$ dV = \pi r^2 \, dy $$

gives $$ V = \pi \int\limits_0^{99} \!\! r(y)^2 dy $$

For the radius we have $$ y = a r^2 + b \iff r^2 = \frac{1}{a}(y-b) $$ and thus

$$ V = \pi \int\limits_0^{99} \frac{1}{a}(y-b) \, dy $$

I leave the integration for you.