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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=e^{x+y} ; \quad 0 \leq x \leq 1,0 \leq y \leq 1

$$

(e-1)^2

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Okay, so we're giving the surface Z equals eat the x plus y and we want to find the area between or the volume between this surface. And this equals zero plane for this region defined to be set of all points, which are, and any coordinated pair, uh, where x is between zero and one, and why is between 01 as well? So this is a little set notation. Basically, what I'm saying is we have our being the set of all points X and y that are cornet pairs r squared. Such that X is between 01 ad wise between 01 as well. Okay. And this low colon means such that, by the way, Um, yeah, So let's get an idea of what we're doing or what our region are Looks like X y. So basically, it should just be our region here, this square region. Okay, um, we want to find the volume between these equals airplane at this surface for this region. So it's basically the double integral. We want to set up a double integral because that's that's what finds us the volume over the region are of her function. Ease the experts y d a. And this is equal Teoh. Let's do a let's do a d y dx integral. It doesn't really matter in this case, But I'm just gonna go with the white. He asks, Um, I'm just gonna rewrite our functional bits each extents each of live by our rules of exponents, um, and says we're doing a d y d x. We want to think about how, why ranges And then how about how X ranges? Because we always evaluate multiple intervals from the inside out. So why ranges from 01 and then X ranges from 0 to 1? So that's just amusing. Way to write the limits for our interval. So now, evaluating this, you have do you want And since we're in a great respect why we think about other every other available as Constance so we can pull this each of excel. Uh, okay, then Now we have each of x times the anti derivative of each, the UAE's US itself. He's a why is there no one? This should result and e minus one, because playing in one into why is just one so each the first tires e. So we have easier and then subtracting what we get when we plug in the lower limit. So each of those you're of power is just one. So we have B minus one and now, playing this into the outside integral, you'll notice a pattern here. So a girlfriend 01 of each x e, my sworn DX were Just don't pull out the constant of E minus one, and you'll notice that it's basically the exact same interval. As this one is, just has a different name set of being why it's using variables of X, so should just result in the same value e minus one. And that means that our double integral or the volume between our two surfaces is e minus one quantity squared, and that is our final answer.

Rutgers, The State University of New Jersey