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Evaluate the double integral over the given region $R$.

$$\iint_{R}\left(6 y^{2}-2 x\right) d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$

14

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Johns Hopkins University

Campbell University

Oregon State University

Harvey Mudd College

we want to evaluate this double integral over the region next ranges from 01 and Y ranges from 0 to 2. Now, since our rectangular region does not depend on the other variable, we could write this interval in whichever way we want the boundaries of it. And in this case, already don't think either them really is easier than the other. Some is going to do d y by d. X so we could go ahead and rewrite this as so 1st 01 So this is going to be the balance for X and in the balance for why will be zeroed to And then we just have six y squared minus two x and we have d Y by D. X because, remember, this is supposed to be ex change and the UAE change. Now we could integrate this. Remember, we first look at this inside integral and we ask ourselves, Well, if X is a constant with respect, why how do we integrate this? So doing that will get the integral from 01 of to why cubed minus two x y. Because remember, X is a constant with respect to y in this case, and then we have DX uh, almost forgot to put the valuation from 0 to 2. Now we have DX dy y just X. Now we evaluate this at two and zero. Remember, this is for where why is equal to our mountains. So we have zero 21 So plug in and two, we should get 16 minus for accident. Plug it in zero. Well, everything just cancels out. So now we can integrate this here with respect X. So that would give us 16 X minus two X squared evaluated from 0 to 1. So that would be 16 minus two. Then when we plug ins or will just get zero, so that would give us 14 or our solution.