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
$\begingroup$
$\endgroup$
1
-
2$\begingroup$ We don't rotate an equation, we rotate a region, but I think I know what you intend. The volume (Method of Cylindrical Shells) is $\int_0^{99}2\pi x f(x)\,dx$, where $f(x)$ is your function. $\endgroup$– André NicolasJun 6, 2015 at 20:20
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
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.
$\begingroup$
$\endgroup$
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.
