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16 Evaluate the iterated integral frcos(y?)dydx by reversing the order of integration: sin(64) A) 3 B) sin(64) cos(64) C) 3 D) cos(64) E) none of the above0 D0 EB0 ...

Question

16 Evaluate the iterated integral frcos(y?)dydx by reversing the order of integration: sin(64) A) 3 B) sin(64) cos(64) C) 3 D) cos(64) E) none of the above0 D0 EB0 A

16 Evaluate the iterated integral frcos(y?)dydx by reversing the order of integration: sin(64) A) 3 B) sin(64) cos(64) C) 3 D) cos(64) E) none of the above 0 D 0 E B 0 A



Answers

$3-14$ Calculate the iterated integral.
$$\int_{0}^{1} \int_{0}^{3} e^{x+3 y} d x d y$$

So Yeah. Okay, so in this problem corner integrating with respect to why first. So that means each of the two X it's just a constant. So to get each of the two X and then the integral of White Cube with respect to y y to the fourth over four from 0 to 4 DX. So you plug in four and you get 64 there. So I'm just gonna put that up front, okay? The integral of each to the two X is each of the two x over to ground 0 to 2. So you get 32 e to the four minus each of the zero. Don't forget to plug in the zero. So 32 years to the fourth, minus one.

Okay. This problem, we're gonna integrate with respect to you first. So either we're gonna have to raise this'll this by no meal to the fourth power, or we're gonna have to do a substitution for that, you plus b squared. So I'm gonna say w is you plus B square. So D w is Do you Okay? So now we have 01 Uh, I forgot this part. Um, if U equals zero than w equals zero plus b squared. So b squared if U equals one than w equals one plus v squared. So now we have the integral B squared to one plus b squared W to the fourth d w and then I still have Don't forget this V here in this devi. So I put vdv. You know what? I don't really like putting that be there. I'm gonna put it over here where it belongs. Right there. So now I was 01 v and I'm gonna integrate this and I get W to the fifth over five from V square to one plus B squared D V. So I have 1/5 01 one plus V squared to the fifth, minus B squared to the fifth and then this V V D v? Yeah. Okay, so now yeah, I have 1/5 01 one plus V squared to the fifth. I'm gonna go ahead and make this into two intervals. So here comes the Vdv Vdv, minus 1/5 in a girl 01 V squared to the fifth, which is V to the 10th. And then B D. V. Okay. I had to give it to both of them. And that's what the square brackets did for me. Can I'm gonna have to make a substitution again. I'm gonna let w equal one plus b squared. Then d w equals to V devi. So I needed to hear which gonna put a one half out in the front here? Yes. If V equals zero w equals one. If V equals one, w equals one plus one square. So too So now I have wanted tint 1 to 2 w to the fifth, D w, minus 1/5 01 B to the 10th times V so v to the 11th d v. So notice on the last one. I didn't change the in points because I did not use uh, you substitution on that 12 totally separate things going on here now. So I have 1/10 w to the sixth over six from 1 to 2. 1/60 to to the sixth, minus one to the sixth. To to the 6248 16 30 64. Okay, so now I have 1/60 64 minus one, which is 63. So 63/60. And then let's see what happens on the other piece of it. And then we'll decide if we need to do some kind of simplifying. So we have here minus 1/5 feet to the 12/12 from 01 So minus 1/60. One to the 12th, minus zero to the 12th. So minus 1/60. And so in the end, minus 1/60 we get 60 to over 60 or 31/30.

In this problem. First time integrating the given value with respect to X, considering via the content so I can write the express and has 0 to 1. He took the power White Cube Access Choir by two 02 y divide, simplifying it further. I cannot evaluate one by two, 0 to 1. Why squad it to the power? Why cube dy now let us take why Cuba is equal to T. Then three wise square dy will become DT going forward and solving it. For that I can write y squared dy is equal to D 80 by three when why is it called to zero? Is equal to zero when y is equal to one he is equal to one cube is equal to one now going forward and simplifying it for the so I can write the expression integration of 0 to 1 Why is required it to the power y cube Divide multiplication one by two on solving it I can ride evaluate one by six Integration of 0 to 1 It is the power t d t on further solving. I can ride evaluate one by six he to the poverty from 0 to 1. On further simplification, I get the value of one by six e to the power one minus e to the power he to the power one minus one, which is equal to one by six. E minus one. This is our final answer.

Grading with respect to why? So I'm gonna move that X. E. To the X. Out here. So we have 12 to 1 of her. Y. Dy D. X. So that gives us the natural log of Y. from 1 to 2. That gives us 021 X. E. To the X. Ln of two minus the L. N. Of one D. X. The Atlanta one is 0. Okay then on the rest of it we're gonna have to do integration by parts. So you as X. DV as each of the X. The U. S. D. X. He is E. To the X. So this is E. X. E. To the X minus the integral of E. To the X. From 0. 1 times the Ellen of two out here. So where you have the l. In of two one each of the one minus each. The one minus zero each of the zero minus E. To the zero. That was clearly cancel That's zero. So I get one times the Ellen of two Natural Log of two.


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