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02. Calculus (Differential Equations & integrals)(a) Find02z when z (x3+y2) = e dx2(b) Find02z when 2 = e(x+y2) axdy(c) Evaluate10x2y3 6 dAWhere D is the region...

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02. Calculus (Differential Equations & integrals)(a) Find02z when z (x3+y2) = e dx2(b) Find02z when 2 = e(x+y2) axdy(c) Evaluate10x2y3 6 dAWhere D is the region bounded by x = ~Zy2 and x = y3 between -2 <y <0y=0x=y? 10x2y3 6 dx dy V=-2 x=-2y2

02. Calculus (Differential Equations & integrals) (a) Find 02z when z (x3+y2) = e dx2 (b) Find 02z when 2 = e(x+y2) axdy (c) Evaluate 10x2y3 6 dA Where D is the region bounded by x = ~Zy2 and x = y3 between -2 <y <0 y=0 x=y? 10x2y3 6 dx dy V=-2 x=-2y2



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Area integrals Consider the following regions $R .$ Use $a$ computer algebra system to evaluate the integrals. a. Sketch the region $R$. b. Evaluate $\iint_{R} d A$ to determine the area of the region. c. Evaluate $\iint_{R} x y d A$$R$ is the region bounded by the ellipse $x^{2} / 18+y^{2} / 36=1$ with $y \leq 4 x / 3$

Okay. It's time to give us the Internet into Rome from 0 to 2 pi over two and then from white. It's a pi over two of the surface. Six times the sign two X minus three Way the ex do you want? Well, let's see. Let me verify. That's correct before anything else. Okay? Written correctly. Now they want us to sketch the region of integration. So let's do that. Are wise from zero Teoh high over to. So we're gonna really getting in a girl from my toe. Love, Rachel, we have a vertical line. X is equal toe high over tea and then we just have a lion ways Equal tanks. So they want us to compete this region. Let's do that. Zero. Hi. So this time I suspect eggs, so we'll end up with negative. Let's see, six over negative three co sign of two x ministry way Evaluated why? And high over to. And let me just verifying that you take the dirt of one that with negative same first negative cancels out and then we won't supplied by a negative three. So the other negative three on the bottom gets comes about only ended with 16. Okay, it looks so now let's evaluate. So get are single and go single veil by variable. Wonderful. See the negative science cancel out. So we have a to co sign? Um, let's see. High minus three away and then weariness of trying to no sign of this time plug in white. We end up with a negative y the way. But Carson is even so, we can just leave that as a bus. And now let's take our so the ant eater to sign of pi minus three way. But we'll need to divide by a negative. They're here. I'm in Over here. It will have a minus. Two sign of why Evaluated at zero on by over two. Okay, lets see. Well, if we value that zeroes in terms definitely zero. And the first term of sign of pie, which is also zero. So the only thing we're concerned with is the pi over two. Let's see what we could get. Negative 2/3 times. Sign of Let's see pi subtract three pine over to Well, I just turns into a negative crying over two, and then we can pull the negative out because it's odd It's a lot on council. A sign of pi over two is just one. We just got a one here on our second term. Will have, Let's see minus two times ones inside of by. Over two. It's one. So end up with two there minus 6/3 when we get negative for over three.

So we're going to be evaluating this integral over a box domain. So we're going to We noticed that we have our bounce X Why? And Z So we're going to be working from the inside to the outside. So first, we're going to be deriving X. So we have a one here. So we're deriving this one with respect to X. So we're leaving these Z and why integral Z out here for now. So those integral those of the anti derivative of one with respect to X is one eggs, and that's going to be from zero to to. That's going to be evaluated from 0 to 2. And we're just we're ignoring these y's and Z's for now and just integrating with respect to X as well you normally would. So leaving those integral is there for now. We're going to plug in two and zero. So we have one times two minus zero times you wide, easy. So that's going to give us negative 222 and then the inner girl from 3 to 6 of to with respect to why and respect tizzy. So now we're going to be integrating with respect, toe. Why because that's the next in most variable. So we're going to be We're going to leave this Z integral out here. For now, I'll evaluate to the integral of two. So that's going to be two times. Why, with respect to why is going to be that anti derivative from the bounds 3 to 6. We're leaving this do you see out here? So now we have the integral of negative 222 plugging in six and three. So we have two times six is 12 minus 32 times three is six easy. We're going to move over to the right side of this paper so that we have negative the integral of negative 222 of six with respect to Z. So now finally, we're going to derive this with respect to Z. So the anti derivative of six with respect to D is going to be is going to be a six time see from negative to positive, too. So we're going to plug in to and negative too. So we have six times two is 12 minus six times negative. Two is negative. 12. So our answer is going to be 24 so If we just had this as a box and we were looking for the volume, our height of the box would be from negative to to to, so that would be a height of four. Our with of the box would be from 3 to 6, so that would be a width of three. And the depth of this box would be from 0 to 2, so that would be a depth of two. So if we had a box that had dimensions of four times three times two, that would have a volume of 24. So that's how we can check in this particular equation when we're deriving.

So we're going to be evaluating this integral over a box domain. So we're going to We noticed that we have our bounce X Why? And Z So we're going to be working from the inside to the outside. So first, we're going to be deriving X. So we have a one here. So we're deriving this one with respect to X. So we're leaving these Z and why integral Z out here for now. So those integral those of the anti derivative of one with respect to X is one eggs, and that's going to be from zero to to. That's going to be evaluated from 0 to 2. And we're just we're ignoring these y's and Z's for now and just integrating with respect to X as well you normally would. So leaving those integral is there for now. We're going to plug in two and zero. So we have one times two minus zero times you wide, easy. So that's going to give us negative 222 and then the inner girl from 3 to 6 of to with respect to why and respect tizzy. So now we're going to be integrating with respect, toe. Why because that's the next in most variable. So we're going to be We're going to leave this Z integral out here. For now, I'll evaluate to the integral of two. So that's going to be two times. Why, with respect to why is going to be that anti derivative from the bounds 3 to 6. We're leaving this do you see out here? So now we have the integral of negative 222 plugging in six and three. So we have two times six is 12 minus 32 times three is six easy. We're going to move over to the right side of this paper so that we have negative the integral of negative 222 of six with respect to Z. So now finally, we're going to derive this with respect to Z. So the anti derivative of six with respect to D is going to be is going to be a six time see from negative to positive, too. So we're going to plug in to and negative too. So we have six times two is 12 minus six times negative. Two is negative. 12. So our answer is going to be 24 so If we just had this as a box and we were looking for the volume, our height of the box would be from negative to to to, so that would be a height of four. Our with of the box would be from 3 to 6, so that would be a width of three. And the depth of this box would be from 0 to 2, so that would be a depth of two. So if we had a box that had dimensions of four times three times two, that would have a volume of 24. So that's how we can check in this particular equation when we're deriving.

Okay, so here we have a region to find by X equals the absolute value of why and X equals two minus y squared. And we want to find the area of this region both with respect, X and disrespectful. I were to do it both ways. We want to make sure that her answer matches at the end. So we know our general form. Yes, Our area is equal to the integral from A to B of F minus G. We're gonna do it for FX, G of X and F of Weijia. A few notes We know that our X equals absolute value of why is actually essentially to separate equations. We have our X equals y in the positive quadrant and we have an X equals negative. Why in the negative quadrant, in terms of X, they're basically the same thing we have like was X And why equals negative X Our parabola X equals two minus y squared. If we want it in terms of X, we get why equals plus or minus the absolute value of two minus X. So now we have our equations in terms of X and Y we're gonna need dollar various points. So we're gonna have potentially these four points. We have the two intersection points and we have to ex intersection point. No, our origin is easy. X equals wide 00 So that's that point. Our parabola is pretty straightforward. Uh, it's gonna be 20 The intersection points. We'll find this one. We have X equals y and X equals two minus y squared. If we set those equal to each other, we get Why equals two minus y squared. You can rearrange it into quadratic equation, and it factors into why. Plus two, Why minus one. So we get why equals negative two and one. The negative two is off our graph. It's not even relevant to the region we're looking at. So are one is important. Plug it in and we get the 0.11 Now we could do that again for the bottom, but we know that both of these air symmetrical around the X axis, so we know that they're going to have the same X value and opposing y values. So we know this is simply one negative one, and that gets us all our our parts. We have the boundaries for the intervals with our various intersection points on. We have all of our equations in terms of X Ally, Let's get started. Let's find our area in terms of X first. So in terms of X, we're going to use a vertical line, Teoh, to check our regions, we're gonna have to regions, you know, divided by the line at X equals one. So our first region is our absolute value and since it's symmetrical, we're just going to use our positive, are positive line and multiply it all by two. So we have to times the Inter girl from zero toe one of X. Yes, plus, her second region is just the parabola. So again, we're just gonna use the positive value and multiply the whole thing by two so it will be one to to and radical. Tu minus X dx are integral for radical two minus X is going to require some use substitution or you is gonna be two minus X and D you is going to be negative. DX then when you resolve that are anti derivative is gonna be two times the quantity of X squared over two from 0 to 1 minus to times two times tu minus X to the three halves all over three from 1 to 2. When we substitute, we're going to end up with two times the quantity of 1/2 minus zero minus two times the quantity of 0/3, minus 2/3. And then that should all simplify down to 7/3. Right? And now, with respect to why, with our horizontal line test, we see that our we're gonna have to regions separated by the X axis. So our first region, our lower region, is gonna go from negative one 20 And with a horizontal line test, the right most equation is our parabolas. That's gonna be our f of X or f of why rather tu minus y squared that will be minus negative. Why or plus Why? Because our g of X is negative. Y do I plus integral from 0 to 1 Once again, our parabola is our f of why and this time RG of y is simply why so it's minus or g a y de y. This doesn't require any use institution. We can just get our anti derivatives right away. So we end up with two. Why minus why cubed over three plus why squared over two from negative 120 plus to I minus y cubed over three minus y squared over to from 0 to 1. Our substitution gets us zero minus the quantity of negative two plus 1/3 plus 1/2 plus the quantity of two minus 1/3 minus 1/2 minus zero. And again, when you simplify everything down, we should come out with 7/3. So the area of this region is 7/3 and that's it. We're done.


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