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172 Points]DETAILSSCALCET8 15.2.007_Evaluate the double integral Fs dA, D = {(x,Y) | 0 < * < 4,0 < y < V}...

Question

172 Points]DETAILSSCALCET8 15.2.007_Evaluate the double integral Fs dA, D = {(x,Y) | 0 < * < 4,0 < y < V}

172 Points] DETAILS SCALCET8 15.2.007_ Evaluate the double integral Fs dA, D = {(x,Y) | 0 < * < 4,0 < y < V}



Answers

Evaluate the double integral.

$ \displaystyle \iint\limits_D e^{-y^2}\ dA $, $ D = \{(x, y) \mid 0 \le y \le 3, 0 \le x \le y \} $

In problem 43. We want toe calculate the double integral for the function. If saying why Divided by oy over the region D we want toe right the x d boy. But what is the first the X or do y Here we want toe evaluate first the integral DX the oy. To get so is the consider disturbed as a constant relative to X. This means we need toe get horizontal segment where X starts at here X equals y. And in this ad, here at this line where X equals two boys, then the first bounds the first limits off the inner integral is from Oy the two boys, while why starts at one and in this ad, too 1 to 2. Let's evaluate the inner integral first integration off a constant multiplied by the X. He's just the constant multiplied by X sign. Why divide by why Multiplied by X integrate from my two toy equals the substitute first by the upper bound the upper limit Science y divided by white multiplied by two y minus saying why divided by why multiplied by why the integral From 1 to 2, we simplify. We have to sign y minus sign. Why? Which is sign why you will. The integration of Sign Roy is just minus cousin Y. We substitute from 1 to 2 equals minus design to minus minus, which is plus co sign one, which is the final answer off our problem.

Okay. So to evaluate the given interval, we first need to draw our ranges. So we're given why equals X squared. So we're gonna draw that. And then we're told we want it bounded from, so we'll start with X equal toe one. So I'm going to draw a line going down or x equal toe one. So obviously you can see and this is gonna be bounded above the X axis. So that means we're looking for this region right here. So with that, we just need to figure out our bounds are ranges. So if we have the double integral of x cosine of why, Okay. So first, I'm going to integrate with respect to y first. So that means I'm taking vertical cross sections. So are vertical cross sections look like this. So that means we're going from the X axis or why you call zero two y equals X squared. So I'm just going to write that up here, and then, as you can see, X is going to go from zero the origin toe one. So X is going to go from 0 to 1 and then so because we're doing with respect to y first and then acts We're gonna have d y times dx. So now all we have to do is just integrate this. So with this, we're gonna have so cosine we're gonna integrate Cosign Y X is just a constant in this case, so we're gonna have integral from 0 to 1 off X. So in a in a girl or anti derivative of cosign, why is sign y? And then we're going to evaluate this from zero to x squared and then times the X then continuing here, we're just gonna plug in X squared and zero. So this is going to give our result of integral of X sign X squared. So if you plug in zero sign of 00 so that term goes away so we're still gonna have D. X and X is going to go from 0 to 1. So now for this one, we're gonna use U substitution. So over here I'm gonna write, use equal to X squared, Do you? Is equal to two x dx, so X DX is gonna be one half d you. So now if we go ahead and, um, plug these in so we're gonna have the integral. So x DX is going to be replaced by one half, do you and then sign X squared is going to be replaced by sign of you. So this is easier now because we know the anti derivative of Sign you. And here are values are still gonna be from 0 to 1 because if you plug in the ranges, X goes from zero. So zero square 201 squared is one. So you goes from 0 to 1. Then from here, we're gonna now just take the anti derivative of sign You, which is negative co sign of you. And then again, you is going to go from 0 to 1 so continuing here, all we have to do now is plug in one and for you and then zero so that this will have negative. Or we're gonna have one half times negative cosine of one plus cosine of zero. And then now we can justify Coastline of Zero is one. So our result here is going to be negative. One half cosine of one plus one half. Yeah,

This problem we wish to give evaluate the given integral. The integral on region G. Of Y squared E. To the X. Y. D. A. Where D. Is the region. Why between zero and four X. Between zero and Y. This question is challenging our understanding of double integral to evaluate the integral. We used to single available integration techniques and step and stuff. When we integrate evaluate the inner integral step to evaluate the outer integral and then step three we saw to evaluate this double integral. Though we have to figure out the the ordering of dx dy dy dx as well as the limits of integration because access between zero and Y two K. One of the bounces functional de actually our inner integral do I then will be our outer integral. Doesn't have integral 0 to 400 yy skirt the X. Y. Z Y X Dy I value first. The inner integral gives why you X Y from zero to Y. This evaluates as integral 0-4. Y. You the White Square do I. Our outer article has the exponential form. It evaluates as one half year to the Y squared from 0 to 4. Plugging in our balance give solution one half of the 16th minus one half

All right. Here's a double inch girl that we're going to evaluate. We want to go from one two and from why minus one toe, one to x plus. Why DX Why? And so we're integrating with respects to ex first. So we're going to get X squared plus x y evaluated from why minus one toe. One of d y. So using FTC When we plug in one, we get one less. Why? Minus why minus one squared. Plus why minus one times Why do I All right, hoops? So, uh, let's, uh, well, we have to do some algebra here. So let's do that Algebra and got one plus why minus why squared minus two plus one plus y squared nine. It's why d y. And now we have to just combine all these like terms. Okay, so that one plus why nine is too y squared. Plus why cubed minus one. I'm sorry. Plus three y minus one. That's just from combining these and distributing the negative so the ones cancel out. And what else do we have? Let's rewrite this into girl very, very nicely. You've got negative too. Why squared? Plus four. Why d y Okay, now we injury. This is a simple integral from calculus one, and then we plug in our pounds. Okay, so we've got negative to over three. Why? Cubed plus two y squared from one to two. So we've got negative too. Over three times, eight plus two times for nine iss. Negative to over three plus two. Okay, so this is ahh when we we, uh, not native to our three time's A plus. Eight plus two over three, minus two. And so we're going to have Well, this is negative. Sixteen over three. Plus two over three plus six. So this is six minus fourteen over three. Um, we can rewrite this just by subtracting fractions. Did you get for over three. Two and we're done.


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