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Follow the step-by-step guide to integrate similar ordered:
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- Use Advanced Options to limit access to the record and set an expiration date.
- Click Save and Close when completed.
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Other Methods Of Ordering On Reverse Side Form
okay so this is going to be another double integral over general regions problem we have to change the order of integration but in this one I'm going to go 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 I'm about to use so again whatever the inside limit of integration is here we're going to integrate with respect to X whatever the innermost limits are I set those limits equal to X because I'm 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 we're 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 that's just a vertical line x equals three okay and then it says on the outside I'm going to graph think about the lines y equals zero so y equals zero and y equals one so y equals zero again it's just the x axis and then y equals one let's 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 you're not sure if it's this region or kind of the top part of the truck the top triangle or the bottom triangle but again since we're integrating with respect to X first I'm going to draw a line that's 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 I'm trapped in the bottom right triangle so that's the region where I'm 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 doesn't have an antiderivative with respect to X so that's why you'd have to go through this whole process that we are now doing ok so let's see if we can't if we can't switch them out ok so again now instead of integrating with respect to X we're 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 we're expressing the Y limits of integration in terms of X now for the x-coordinates again I think over that region what's 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 we've switched out the limits of integration and we simply need to integrate this okay so let's do that real quick okay so I'm going to integrate with respect to Y first that's the innermost part which is the good thing because we can actually do that now so I'm 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 you're 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 I'm plugging in y equals 1/3 X so I'm 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 you'll 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 we're almost we're almost there ok so getting a little bit better I hope so I'm going to factor the 1/3 out front I like to just pull my constants out front I'm integrating from zero to three and then I'm going to write the X first e to the x squared DX and now to integrate x times e to the x squared so it's actually good we've got that that extra we've 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 let's see so our 1/3 is going to be out front the integral let's see so we know x squared that's what we're calling you so we're going to have e to the u that'll 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 we'll get 1/2 D U is equal to X DX so I'm 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 we'll 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 we'll get you equals zero squared or our lower limit of integration will turn into zero since we're 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 we're 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 we'll 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 it's a little more general well it's 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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