Point) Evaluate Zet+y dV where B is the box determined by 0 < x <4,0 < y < 5,and 0 < 2 < 3. The value is...

Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide $ B $ into eight sub- boxes of equal size.

$ \iiint_B \cos (xyz)\ dV $, where
$ B = \{(x, y, z) \mid 0 \le x \le 1, 0 \le y \le 1, 0 \le z \le 1 \} $

So we have to first set up our our triple injury. Can sar expound You're going from zero to three. Are y Ballenger learning from zero tax and r z bounds. We're going from negative X minus Y x plus y t y. So again I'm doing d z do you? Why tx So first doing with respects to Z we're just going to get why times see as are integral that the wise the ex Okay, so after we put these Belgian, we're going to get Why Times X plus y Linus. Why? Times x minus y do you live? D x so nice Cancellation happens. Well, I get Why squared d y, Jax? Okay. Integrating that we're going to get Why cubed I looted from zero X uh, D x. That one third is up here now. Two thirds of to go from zero two, three of just Sorry, which me of ext X cubed Pontiacs. So you get two thirds times one fourth x two four zero three Uh, we'LL end up with one six times three to the fourth. It's one six times eighty one and we can let's see if we can divide this stuff here. So if we divide the top and bottom by three. Twenty seven over too

It's a valuable singer girl and first will integrate with respects Teo X Then why don't you see? Okay, so the x Y Z squared So these are the expounds thes air the Y browns and the are the bounds. Okay, So integrating with respects to ex first X over two squared Why plus X Z squared from zero to two d wide, easy. So we're just going toe kind of go through this. It's very it's not too difficult calculation, but we're just going to show that the integration order doesn't really matter. All right, So when we plug in two, we're gonna get to squared over two witches for over two, which is just too. Why plus Tuesay in a new plug in zero, everything disappears. So that works out very nicely. Uh, now integrating with respects toe. Why that Why squared? Plus two z squared. Why? From zero to one. Easy plugging in again. We get really nice results. We just got one plus two z um squared, right? I'm unplugging Julius. You're all right. So now when we plug in through integrate with respects to Z, we get the following from zero to three. So you got three plus two thirds times. Um, let's see here. Twenty seven two three. That's eighteen, which is twenty one. Okay, so now let's go ahead and switch these orders of integration. So we're into Grant is still the same. So do personal Do X dens e then wide. Thinks we're still in during the zero to two z's range from zero to three and the wise range from zero to one. Okay, so we already did this first integration. So this first integration was already did. So I'm gonna skip it in the sense that we already know the answer. Right? That circle and green is just this guy here in Greek. Okay. Oops. So, that guy in green just to lie plus two z z square right now. If you're busy, then do you want s O That too? Why Z plus two thirds Z cube zero three fti line. So you're the one. Get six y plus. Let's see here. Eighteen six, lifeless Tier six y plus a team. Do you want Okay in generating once more. Three y squared was eighteen. Why? I'm sure the one and that's twenty one. Okay, so let's change the order Tex Why he squared So let's do it easy. Uh, d x d y so Z's go from zero to three. The excess from zero to on the wise from zero. Yeah, so zero one Your team x y z z cubed over three d x d wine shooting from his fear of the three zero one jury, too. Three X y plus, Let's see here twenty seven Divided by three That's going to me nine d x d y. So that three over to X squared. Why close nine x zero King do you like? So that's six wine plus eighteen. Do you know why? And once again, it's the same thing.

So let's go ahead and set up this integral Stop starting this triple integral King E to Z over. Why go d Z? They're the DX then? Then we go D y all right. So very simple integration here. The integral of this with respects to see is why e z over. Why? From zero? Yeah. Two x y d x the wine. So when we plug in X Y, we're gonna get why to the ex mine is why ducks Do I. All right, Integrating once again the respects tow x this time e wh I i e to the x minus x y from one from Weida one Do you want? It's why he to the y minus. Um, let's go back here. And we just, uh ah. Minus. Why minus Why, um keen to the why I'm sorry. Uh, we're plugging in one here, so plugging in one here, you know, here, we're plugging in. Okay, so cleaning this up just a just a tad bit. You minus one. Why? Minus y e to the y plus y squared Do you want? Okay, so some of this integration we could do quite easily just off the top of our heads, but this guy's going to require integration by parts. Okay, so let me do this Integration. Second, if we do this integration first. Why the minus one? Even eyes one. Why squared over to plus why? Cubed over three from zero to one minus. I'Ll do this integration by parts in red. Very standard integration by parts, isn't it? Isn't that why eat the Y from zero to one minus into girl from one toe, zero e? Why d why all right, so kind of finishing this up Slowly but surely we get this over too. Plus one over three. Minus me. Um, might this be minus, um, Dennis Egan. So you just did a slowly That's the first thing to go minus D to the y from one to zero. Okay, so, uh, just gone to the next page e minus e my It's one e minus one over to bliss wasn't there. Plus one

All right. So we wantto go back to the beginning, so we want to use the midpoint rule for triple in jungles. And that's made this into go. So I would have drawn. Here is the X and Y and Excellency boxes. So you can see this will give us eight points. Real pictures in three dimensions and maybe a little confusing to figure out what we split this block box in half. And then we split each face in half. A swell could, so we got eight points, okay? Hate sandboxes for his playing. And then this is how we're going to figure out our volume. So we got our delta volume not too hard to figure out. Delta six Delta y Delta Z. That's just what volume is. Turns out to be one, and then we find each one of these points, right? You got eight of them. Plug all this stuff into a calculator and get our answer