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So this is going to be another double integral over general regions problem we have have to change the order of integration. But in this one. Im going to a little further the other example.
I simply switched out the limits of integration and this one i want to you know work the problem out to completion. So okay. So you may want to look at that the other one though because i think i talked a little bit more about the process.
Im about to use so again whatever the inside limit of integration is here were going to integrate with respect to x. Whatever. The innermost limits are i set those limits equal to x.
Because im going to graph those those two lines again in this case. So. If you think about x.
Equals. 3y. Well we could certainly multiply.
Both sides by one third. And have one third x. Equals.
Y. So were just going to have again a line that goes to the origin of slope. One.
I like to label as i go y equals. One third x. And then it says.
The other line. That i have to graph is the line x. Equals three.
So thats just a vertical line x. Equals. Three okay and then it says on the outside.
Im going to graph think about the lines y equals zero so y equals zero and y equals one so y equals zero again its just the x axis and then y equals one lets see if you plug in the x coordinate of three notice. We this line would intersect. We would get 1 3.
Times. 3. Or sit right here.
At one. So the line y equals. 1.
Is going to go through the intersection of those those. Two red lines. Ok.
So again. You know maybe it would be a little confusing maybe youre not sure if its this region or kind of the top part of the truck. The top triangle or the bottom triangle.
But again since were integrating with respect to x first. Im going to draw a line thats parallel to the x axis. It says.
The leftmost curve should be the line x. Equals 3y. Which would be this line.
And it says. The rightmost curve would be the line x equals. 3.
So im trapped in the bottom right triangle. So thats the region where im going to be integrating okay again you know if you could integrate this straight forward by all means do it. But the problem is e to the x squared doesnt have an antiderivative with respect to x.
So thats why youd have to go through this whole process that we are now doing ok so lets see if we cant if we cant switch them out ok so again now instead of integrating with respect to x. Were going to switch it out and integrate with respect to y. So things are going to change a little bit so i want to make it dy first and then dx afterwards okay so if i integrate with respect to y the bottom curve you would be hitting would be the line y equals zero.
So there will be our lower limit of integration then the top curve that i would be hitting would be the line equals one third x. And again remember you want to express it in terms of the other variable. So were expressing the y limits of integration in terms of x.
Now for the x coordinates. Again. I think over that region.
Whats the smallest x. Coordinate that i use well the smallest x coordinate. I use would be zero.
But the largest x coordinate that gets used would be three so now weve switched out the limits of integration and we simply need to integrate this okay. So lets do that real quick okay so im going to integrate with respect to y. First thats the innermost part.
Which is the good thing because we can actually do that now so im going to have the integral from 0 to 3. When i integrate e to the x squared with respect to y. Again you can treat x.
Like a constant so e. To the x squared. Would just be a constant whoa excuse.
Me so if you integrate a constant with respect to y. Youre just going to simply get that constant times y and i like to remind myself. This is from y equals 0.
To y equals 1 3 x sorry wonderful austan allergies are killing me and then we have to integrate with respect to x on the outside ok so let me evaluate this 0. To. 3.
Ok. So im plugging in y equals. 1 3.
X. So. Im going to get e to the x squared.
Times. 1 3. X.
And then the lower limit of integration well when you just plug in y equals. 0. Youll have e to the x squared times 0.
Which is simply going to cancel out the lower limit of integration and make it 0. Dx. So now we simply have to integrate this and were almost were almost there ok so getting a little bit better.
I hope so im going to factor. The 1 3. Out front.
I like to just pull my constants out front. Im integrating from zero to three and then im going to write the x first e. To the x.
Squared. Dx. And now to integrate x times.
E. To the x squared. So its actually good.
Weve got that that extra weve picked up this extra x term. Because now we can just do a u substitution will let u equal x. Squared.
The derivative will be 2x dx. Okay. So lets see so our 1 3.
Is going to be out front the integral. Lets see so we know x squared. Thats what were calling you so were going to have e to the u.
Thatll take care of my e to the x squared part. We still need to replace x. Dx well.
We know d u. Is equivalent to 2x dx. So if we simply multiply both sides by 1 2.
Well get 1 2 d u. Is equal to x dx so im going to have 1 2 d u. In there and remember when you have a definite integral that you do a u substitution.
On you have to change your limits of integration. So that just goes simply back to our original u. Substitution.
So so we look at u. Equals. X.
Squared. So. The upper limit of integration was when x equals.
3. So well get u. Equals.
3. Squared or 9. So the new upper limit of integration will become.
9. The lower limit of integration was 0. So likewise when we plug that into our substitution.
Well get you equals zero squared or our lower limit of integration will turn into zero since. Were multiplying. We can factor the 1 2.
Out 1 3. Times 1 2. Is going to give us one six human colors going on here that will be 1 6.
So we pull the 1 2. Out. And now were simply integrating e to the ud.
You almost there we get one sixth. The integral of e to the u is e to the u from 0. To 9 plug in your limits of integration.
Well get 1 6. Times e. To the 9 minus e.
To the 0. Remember e. To the 0.
Is simply. 1. So we finally calculated our integral one sixth e to the 9th minus 1.
All right i hope this example makes some sense if this is the first one like i said you may want to look at the other its a little more general well its just another example of setting it up hopefully. The integrations not too bad. But if you have questions feel free to post them hopefully either me or somebody else can can help point you in the right direction.
So all right. I hope this makes some sense and good luck out there .
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