The most classic trick of solving a double integral

Added: 2026-05-12

[00:00:00]One of the most important techniques when we are doing double integrals is to be able to change the water of the differentials. And this is also a very common test questions that you have to know for your calculus three class.

[00:00:12]Let's have a look at this example.

[00:00:14]On the inside we have e to the x to the fourth power. Unfortunately, we cannot integrate this function in the x world.

[00:00:20]It does not have elementary answers.

[00:00:23]So, let's try change the water of the differentials. Meaning we want to integrate this in the y world first. And to make that happen, we will have to change this bounds.

[00:00:34]And let me show you guys the quick way to do this right here.

[00:00:37]So, we are in the x world first. That means x is going from here to here. I can put this down as an inequality.

[00:00:45]x is in between of the cube root of y and two.

[00:00:51]And notice this time we already know that y is from zero to eight. So, we can say this right here is greater than or equal to zero.

[00:01:00]And the reason we want to do that is because now I want to raise everything to the third power.

[00:01:07]And of course, because everything is non-negative, when I raise everything to the third power, the inequalities stay.

[00:01:13]So, here we just get y and then we get x to the third power and that will be eight. And of course, technically you also raise that to a third power, but that is just still going to be zero anyways.

[00:01:26]Good. In fact, we can figure out what the bounds are already.

[00:01:30]And later I'm going to draw pictures for you guys to make it more clear. But, that's it.

[00:01:35]e to the x to the fourth power dy dx In the y world, here's the y.

[00:01:44]Remember, we look at the bottom function to the top function.

[00:01:48]The bottom function is zero. The top function is x to the third power.

[00:01:54]Done.

[00:01:55]And then once we get to the x, it's a left most number to the right most number.

[00:02:01]Well, you can look at this and that and then just isolate the x, but you can look back. x is in between of zero and two. So, zero and two.

[00:02:10]Same thing. And then you're done.

[00:02:11]Seriously, just like that.

[00:02:13]Now, here's just a quick picture for you guys.

[00:02:17]First, when x is going from cube root of y, imagine if this is equal to x. That means when I cube both sides, I get y is equal to x cubed.

[00:02:29]So, it goes like this.

[00:02:32]So, this is technically our function.

[00:02:35]Cube root of y is equal to x.

[00:02:38]So, here it's going to be a left most function because we're in the x world first.

[00:02:44]And then we are going to go up to So, two is somewhere right here.

[00:02:50]So, that will be my right most function.

[00:02:53]So, we are looking from here to here.

[00:02:55]And then y goes from zero to eight.

[00:02:58]So, the bottom number up to the top number, which is eight.

[00:03:02]So, we're talking about this region here.

[00:03:07]So, again, left most function to the right most function, bottom number to top number, dx dy.

[00:03:14]Now, if I'm looking at dy dx, we are looking at the bottom function, which is x which is y is equal to zero.

[00:03:23]This is y equal to zero.

[00:03:26]And then the top function is this part.

[00:03:29]Which we say that is y equals x cubed.

[00:03:33]Let me just write it down right here.

[00:03:37]And then once we are in the x world, we look at the left most number, which is zero, and then the right most number, which is two.

[00:03:47]Again, we are still talking about this region. So, just like that.

[00:03:51]Now, once we have all this, we can just go ahead and integrate it because integrating [clears throat] a constant in the y world is super super easy. It's just going to be y times that.

[00:04:00]So, we get zero to two. We get y e to the x to the four. And we plug in zero and x to the third power dx.

[00:04:13]Plugging x to the third power here, we get exactly just that.

[00:04:21]Plugging zero, it's gone. So, that's very nice. So, that's what we have to do and then dx.

[00:04:25]And then right here, let's just go ahead and do a very nice u sub. Let u equal x to the fourth power. du is equal to 4x cubed dx.

[00:04:38]We need a four. Let's multiply by the four and then divide it by it right here.

[00:04:43]So, we have 1 over 4.

[00:04:45]This and that is our du.

[00:04:50]And then we have e to the u.

[00:04:53]And let's just finish everything in the u world.

[00:04:56]When x is equal to zero, we can see that u will be going from zero.

[00:05:01]And when x is equal to two, two to the fourth power 16. So, u will be going to 16.

[00:05:07]Okay.

[00:05:08]Integrating e to the u is just e to the u. So, leave it like this. e to the u.

[00:05:14]Uh I will still show you guys all the steps. e to the u and then we go from zero to 16.

[00:05:20]I'm plugging 16, plugging zero. 1 over 4 e to the 16 minus 1 over 4 e to the zero's power is one. Right?

[00:05:32]And that's pretty much it. I'm going to leave it like this.

[00:05:35]Yeah.

[00:05:36]Just like that. That's it.