# Volume of the frustrum is incorrect if you derive it!!!!

• Dec 6th 2012, 01:53 PM
Drigomaniac
Volume of the frustrum is incorrect if you derive it!!!!
So most likely I am the one with a mistake...but hell I can't get it to work.
I am attaching my work. So I try to derive the volume of a cylinder from the area of the circle (A(circle) = pi*r^2)....from h=0 to h=H.
Same with the cone...but here we know that the radius depends on the h so A(circle_for_the_cone)= A(h) = (R/H)*h, you derive it and gold.

So I did the same with the cone but changing the integral limits....and doom! Cant get it to work. Could somebody help me out using this methodology???

The volume of the frustrum should be: V (frustrum)= pi*h/3*(R1^2+R1*R2+R2^2)

Check attached picture! :)
Thanks!
• Dec 6th 2012, 04:51 PM
skeeter
Re: Volume of the frustrum is incorrect if you derive it!!!!
see attached sketch for my variable labels ...

equation of line in quad I is $\displaystyle y = \frac{r}{h} \cdot x$

volume of small cone ...

$\displaystyle V = \pi \int_0^h \left(\frac{r}{h} \cdot x\right)^2 \, dx$

$\displaystyle V = \frac{r^2 \pi}{h^2} \int_0^h x^2 \, dx$

$\displaystyle V = \frac{r^2 \pi}{h^2} \left[\frac{x^3}{3}\right]_0^h$

$\displaystyle V = \frac{r^2 \pi}{h^2} \cdot \frac{h^3}{3}$

$\displaystyle V = \frac{\pi r^2 h}{3}$

frustrum volume = volume of large cone - volume of small cone

$\displaystyle V = \frac{\pi R^2(H+h)}{3} - \frac{\pi r^2 h}{3}$

$\displaystyle V = \frac{\pi}{3} \left[R^2(H+h) - r^2 h \right]$

$\displaystyle V = \frac{\pi}{3} \left(R^2H +R^2h - r^2 h \right)$

$\displaystyle V = \frac{\pi}{3} \left[R^2H + h(R^2 - r^2) \right]$

using similar triangles, $\displaystyle h = \frac{rH}{R-r}$

$\displaystyle V = \frac{\pi}{3} \left[R^2H + \frac{rH}{R-r} \cdot (R^2 - r^2) \right]$

$\displaystyle V = \frac{\pi}{3} \left[R^2H + rH(R+r) \right]$

$\displaystyle V = \frac{\pi H}{3} \left(R^2 + Rr + r^2 \right)$
• Dec 6th 2012, 10:23 PM
Drigomaniac
Re: Volume of the frustrum is incorrect if you derive it!!!!
Tried the same approach but how do you get h to be in terms of the other variables...this might be a stupid question...but i will appreciate if you can help me.

h= \frac{rH}{R-r}

Also instead of doing large cone - small cone, could we get the solution deriving the area of the frustrum as I have it in the attached document?

Thanks!
• Dec 6th 2012, 10:49 PM
MarkFL
Re: Volume of the frustrum is incorrect if you derive it!!!!
If I were going to derive the volume of the frustum of a cone using slicing, I would let the volume of an arbitrary slice be:

$\displaystyle dV=\pi r^2\,dx$

With $\displaystyle r_1$ as the radius of the top of the frustum and $\displaystyle r_2$ as the radius of the base, we may let the radius of the slice at $\displaystyle x$ be:

$\displaystyle r=\frac{r_2-r_1}{h}x+r_1$

Hence:

$\displaystyle dV=\pi \left(\frac{r_2-r_1}{h}x+r_1 \right)^2\,dx$

And so the volume is:

$\displaystyle V=\pi\int_0^h \left(\frac{r_2-r_1}{h}x+r_1 \right)^2\,dx=$

$\displaystyle \pi\int_0^h \left(\frac{r_2-r_1}{h}\right)^2x^2+\frac{2r_1(r_2-r_1)}{h}x+r_1^2\,dx=$

$\displaystyle \pi\left[\left(\frac{r_2-r_1}{h}\right)^2\cdot\frac{x^3}{3}+\frac{2r_1(r_2-r_1)}{h}\cdot\frac{x^2}{2}+r_1^2x \right]_0^h=$

$\displaystyle \pi\left(\left(\frac{r_2-r_1}{h}\right)^2\cdot\frac{h^3}{3}+\frac{2r_1(r_2-r_1)}{h}\cdot\frac{h^2}{2}+r_1^2h \right)=$

$\displaystyle \frac{\pi h}{3}\left((r_2-r_1)^2+3r_1(r_2-r_1)+3r_1^2 \right)=$

$\displaystyle \frac{\pi h}{3}\left(r_2^2-2r_1r_2+r_1^2+3r_1r_2-3r_1^2+3r_1^2 \right)=$

$\displaystyle \frac{\pi h}{3}\left(r_2^2+r_1r_2+r_1^2 \right)$
• Dec 7th 2012, 03:10 AM
skeeter
Re: Volume of the frustrum is incorrect if you derive it!!!!
Quote:

Originally Posted by Drigomaniac
Tried the same approach but how do you get h to be in terms of the other variables...this might be a stupid question...but i will appreciate if you can help me.

$\displaystyle h= \frac{rH}{R-r}$

similar triangles ... sides are proportional

$\displaystyle \frac{h}{r} = \frac{h+H}{R}$

solve for $\displaystyle h$
• Dec 7th 2012, 08:53 AM
Drigomaniac
Re: Volume of the frustrum is incorrect if you derive it!!!!
Great! Got it....thanks again for all your help!