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Problem 29 Hard Difficulty

Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.
$$
z=8 x+4 y+10 ;-1 \leq x \leq 1,0 \leq y \leq 3
$$

Answer

$\iint_{R} f(x, y) d x d y=96$

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Video Transcript

All right. So given the surface, um, z equals eight X plus four y plus 10. Um, we want to find the volume between fire to draw this out a little bit. Uh, just make sure the street perfect. So this is our X y z cling. You want to find, um, the volume between the Z equals zero plane, which is basically this plane here and the surface If we graphed out, um, this surface would just be a plane, but I'd rather not go through graphing it because it could be a little time consuming. So it's basically just the volume between this and this plane that we want to find. And we do this by evaluating a double integral and the double integral one set up. It's basically the double mineral over region are and region are is defined as when X is between. Basically, R is equal to the set of all points such that X is between one second, one on one on wise between through and three. So this is a little set notation here. Basically, we're saying that the region are is thes set of all points x and y such that exes between negative one and one and wise between zero and three. So going back to this setting up this double integral were saying the double integral of region are of our function. Eight X plus four y plus 10 d A. Um, now integrating respect to why first our experts doesn't really matter. It's they're both pretty easy. But for the six of the video, I'm going to integrate aspect to y First, we're gonna have a d Y DX interval. If you choose Teoh, do with dx dy y integral. It could be a little exercise for you. Teoh. Verify that either way is, uh, correct. It's just you may have different limits in different positions and you'll see in a sec. So have you ever function there? Basically, we're finding the volume between the Z equals zero plane and this function over a specified region. So how I like to get our limits in the right places is I think about the fact that we're always about being into girls from inside out. So when we're integrated, spent two wife furs, we want to think about how it is. Why range in this problem and why ranges from 0 to 3. So we have dilemmas from 0 to 3, and then we want to think about the outside new role. And how does that very well range? Basically, yeah. Um, and since it's x X ranges from negative 1 to 1, so this would be our double integral that we need to evaluate. And this would give us the volume between the Z equals zero plane and our surface eight X plus four y plus 10. So, evaluating this, we observed three of eight X plus four y plus 10. Do you? Why, um, evaluating this we have e x y because eight X is a constant. In this case, we're integrating spent supply. We have eight x y plus four y squared over two plus 10. Why, um, evaluates from 0 to 3 and we're gonna let this e it's gonna be some flags. Eight x y plus two y squared plus 10 y 0 to 3. So bloody in the top limit. We have, uh, okay, since we're integrating spent two. Why remember, it's a plug in tow. Why, that's how I kind of remember which variable we need a plug into. So plug ins or why we have 24 x plus. Why squares 09 times two is 18 plus 30 then, um, flooding the second or the lower limit We get zero. So now we have to plug this back into our outer Israel So that a girl from Nick Oneto 1 24 x plus 40 e t x. This should just be equal to that role for negative should be equal to the reverse power rule. Here 40 s once one then playing all this end for me just a little bit more. There should be equal to plug in one. We get 12 plus 48 then minus when we plug in one, get 12 minus 48. So we have 12 minus 12 plus 48 plus 40 and this should be equal to 96. That's our final answer.

Rutgers, The State University of New Jersey
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