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Evaluate each double integral.

$$

\int_{0}^{2} \int_{0}^{3 y}\left(x^{2}+y\right) d x d y

$$

44

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Missouri State University

Campbell University

University of Nottingham

Idaho State University

Okay, so we're given this double Integral, um, enrolled from Syria to time four of the interval from sort of three. Why of X squared Plus y tst wife. We would just want to evaluate this. Uh, so let's do the inside first. Uh, since we're integrating spent two X, we think about other. Every other variable is a constant. So when we're integrating, we have absolute 3/3, Um, by the reverse power rule on then plus x times y Because why is this our constant in this case? Or if you are too, draw a parallel, be wide times x wiser, constant. And then from 0 to 3. Why? And I want to know what variable we need a plug in these limits to We just think about what bearable were integrated respect to right now, which is just actually plug in tow. X So we have three y to the third over three plus three. Why squared minus plugging in two X, We get zero plus sirrah. So now this should just result in nine. Why? To 3rd 3 y squared and then we have to put this into the outside angle. So it's single from 0 to 2 of nine wider third plus three y squared d y. Then, evaluating this, we use the reverse spiral again. Nine y to the fourth over four plus three wide with third over. Three started to the sequel to this just simplifying a little bit and then plug in the top limit. We have nine times 16/4 plus eight, then should be just minus zero. Because when we plug in zero into both here and here should just be zero. Then we have nine times four plus a, and this result in 44 that is our final answer.

Rutgers, The State University of New Jersey