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/sci/ - Science & Math

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6147329 No.6147329 [Reply] [Original]

Alright, /sci/

I'm working through some basic triple integrals - stuff like volumes between cones and hemispheres. Setting up these integrals usually isn't too bad, but almost always ends up placing at least one or two square root functions in the integrand.

Now, I know that we have some techniques to deal with this. Trig subs come to mind, but usually end up messy if they're in the middle integral.

I'm also aware that converting them to cylindrical or spherical coordinates can clean things up, but the section I'm working from now only deals with cartesian coordinates. And yet, they are able to pull exact answers out of these things.

Is there something major I'm missing here that simplifies these kinds of integrals?

>> No.6147343

sqrt(x) = (x)^(1/2)

>> No.6147357

>>6147343
True, but you usually end up with multiple variables squared under a square root, like sqrt(8-x^2-y^2) for example, if you're dealing with a hyperbolic cylinder of some sort in 3 space.

And they're not usually simple U-subs, either - there's generally not an x outside the square root to cancel.

But these questions are among the first problems in the section. They don't explain what they're doing in the chapter or the solutions, but end up with exact answers.

>> No.6147358

>>6147329
http://www.wolframalpha.com/input/?i=int+-1+to+1+int+-sqrt%281-x^2%29+to+sqrt%281-x^2%29+1+dydx

http://www.wolframalpha.com/input/?i=int+0+to+2pi+int+0+to+pi%2F2+%28int+0+to+1+%28p^2*sin%28phi%29%29+dp%29+dphi+dtheta

I just did your homework for you. Give me money.

>> No.6147359

>>6147357
Use a different coordinate system

>> No.6147364

>>6147358
Well, it's more studying than homework, but thanks.

But, in reference to this, and to >>6147359
, this is actually in a section before coordinate transforms are introduced. I can see that it's be easier to just switch over the cylindrical/spherical coordinates, but they're doing everything in cartesian.

What I'm really trying to figure out is what technique they're using to make these cartesian integrals tractable, just in case converting coordinates doesn't help and/or only complicates things.