Triple Integral Calculator Step by Steps

Video Transcript

Hello again everyone, this is Tom from EveryStepCalculus.com and EveryStepPhysics.com. We’re gonna do a triple integral from Calculus 3 right now.

This is an example of Patrick JMT, my favorite instructor on the Internet and youtube. So I’m gonna show you how it works on my TI-89 program. I don’t know anybody can do that problem, he can do it because he’s a genius, but for us students, etcetera, how do we do it?

So let’s get started. Index8() to get to my menu. I’m gonna scroll up because I can go to the bottom of the menu then, instead of going down quicker to the T section and we’re gonna choose triple integral. And we’re gonna enter our function, you have to press alpha before you enter anything into these entry lines here in my programs, okay?

Alpha X times sin of y. I always show you what you’ve entered you can change it if you want. And we’re gonna use the order of integration, which is dx, dz, dy, which is in the example. You have the other choices in case that’s given on a test also. region cue enter these.

And we’re gonna enter the region, q. We’re gonna enter these limits. This is alpha 0 for the x one. Alpha square root of 4 minus Z squared. Made a mistake so I gotta go back. Choose number 2. Alpha 0. Alpha square root of 4 minus z So here’s what you write on your paper That’s better, say it’s okay.

Next one for the y is alpha 0, alpha pi. That looks okay. and alpha 0 for the z. alpha 2. That’s okay. So here’s what you write on your paper, the way you write a triple integral with dx, dz, dy order of integration. Here’s the function in here. We’re gonna do the dx first, and you put this over here with these lines, when you’re doing the range over this integration here.

And here’s the integral of the first here. Of the first function. And if x equals the upper range… I show quotation marks here but you put parenthesis in there because you’re substituting this amount for any X in the integral, and it equals this, minus sign etc.

And then we do the lower integral. X equals zero, and there’s 0 plugged in, you put parentheses around this instead of quotation marks, okay. And here’s the answer, upper range minus the lower range equals this right here. So that becomes the new integration function and I show you that here. Dz, dy is left, okay?

So now we integrate that, come up with this, minus sin z, z squared, et cetera and over this range here, 0, 2. Add z equals 2. Here’s the answer here, at z equals 0 plus these in for all the z’s in the problem. And the answer is this, the upper range minus the lower range is 8 sin y divided by 3.

We’re gonna use that for the integration function with the range of 0 and pi. At y equals pi minus 8 cosine, here’s the 8/3. Y equals zero, you plug that in here, you get minus 8/3. Upper range minus the lower range, notice the minus times the minus, you can’t remember that stuff a lot of times. Turns out to be, the volume is 16/3. Okay?

No problem. So go to my site, subscribe so you can see other videos I might make. Or you can go to the menu on my main site and go scroll down to what you need to learn. And see my program works, because it sure teaches you quicker than a book or anything else. Okay, so have a good one.