order of integration in triple integral

[mcwang719]

Hello. I had a question dealing with triple integration. Here's the problem:

Consider a solid cone. The radius of its circular base is 2 and its height is 2. If the density function is $\displaystyle \sigma (x,y,z) = x^2 z$ find the mass.

So $\displaystyle z = \sqrt {x^2 + y^2 } = r$ and $\displaystyle x^2 = r^2 \cos ^2 \theta$ I understand the limits of integration. My question is the order of integration, I know I have to use dz r dr d(theta). Does it matter which order to use and, if it does matter, how do I know the order to use?

 

[galactus]

Try $\displaystyle r\ dz\ dr\ d{\theta}$. This is the order most commonly used.

\(\displaystyle \L\\\int\int\int(f(x,y,z))=\int\int\int{f(rcos{\theta},rsin{\theta},z)rdzdrd{\theta}}\)

 

[mcwang719]

But how do we know which order to use?? thanks!

 

[tkhunny]

Once in your lifetime, you should do it all six ways, just to prove that it can be done. The answer is unique. If you set it up and perform it correctly, you will get the same result.

 

[galactus]

Is this what you got for your integral?.

\(\displaystyle \L\\\int_{0}^{2{\pi}}\int_{0}^{2}\int_{r}^{2}r^{3}cos^{2}({\theta})dzdrd{\theta}\)

Check me out.

Try rearranging the limits as tkh said, and see if you can arrive at the same answer. Try $\displaystyle drdzd{\theta}$, for instance.

\(\displaystyle \L\\\int_{0}^{2\pi}\int_{0}^{2}\int_{0}^{z}(r^{3}cos^{2}({\theta}))drdzd{\theta}\)

See how z isn't necessarily r to 2 as before?.