Stuck on Integration

evolving_doors

I've been working through triple integral for a sphere ( by cartesian) and I've gotten past the 1st stage. Now I'm trying to integrate sqrt(9 -y^2 -z^2 ) dy but getting nowhere fast... any hints?

I've seen one solution as (ysqrt(9 - y^2 - z^2) + (9 - z^2) arcsin(y/sqrt(9 - z^2) but havn't a clue how he made that leap!:(

Thanks

ZorbaTehZ

Your question is too vague, what exactly do you want to integrate. And if it involves a sphere then there is no reason not to make a change to polar coordinates before integrating you only make your life more difficult not doing that.

evolving_doors

Yes polar would be a lot easier, however the evaluation path I'm on is by using cartesian coordinates only.

If it makes any difference the y limits are -(sqrt9 -z^2) to +(sqrt9 -z^2) and after that I'll be looking for dz with z limits from -3 to 3

evolving_doors

or maybe to put it another way... how is this step made?

Integration.jpg

ZorbaTehZ

That step is not trivial and I suspect he/she already explained it in a previous example (based on the 'note').

It amounts to how might one integrate [latex]\sqrt{c-y^2}[/latex] which is usually carried out by a trigonometric substitution see here

evolving_doors

Ok thanks, I subbed in (sqrt(9-z^2))sin(theta) instead of y and changed the limits etc. All worked out then:pac: