00:01
Welcome to this lesson.
00:02
In this lesson we would calculate the double integral for an area of our.
00:23
For this one it's very convenient to start with the y first and finish with the x.
00:45
So we take the whole of this because it has become differentiation by paths.
00:53
So we will let you equals to to the power x y then we differentiate you respect to x and that would be this one because we haven't why would differentiate with respect to y so if you do that this becomes x e x y d y okay but this is what we have as the u so you have the u equals to x u y we can divide by u x ux u x so they have the y as d u over x u okay so we can write the whole thing again before we do that's let's look at the limits so we have the when x when y is called to 0, the whole of this becomes 0.
02:24
So we have e to the power 0 that is 1.
02:30
And we have the upper limit when y is equal to 1.
02:36
We have e to the power x.
02:40
So from 0 to 2, the inner integral becomes 1 to the inner integral becomes 1 to the power x.
02:54
X so in place of e u we'll put in you there then in place of the y will put in the u we'll put in the u all over x u the x is also there so that this so cancel that's and you have 0 2 1 e to the power x the u have the u of the x out so this becomes u that start from 1 to x x this becomes that for e x minus 1 x okay so this when we integrate this we have e x minus x from 0 to 2 .2 so if we insert set we have e to the power 2 minus 2 then when we put in the 0 we have negative e to the power 0 plus 0 okay so this becomes e to the power 2 minus 2 minus 1 okay so this becomes e to the power 2 minus 3 okay the next one that we will look at is the b part we have the integral from 0 to 4.
05:53
So integral from 1 to e have 6x, then y squared all over y, then you have the y, the x.
06:11
Okay.
06:15
For this one we'll start with y, then we'll start with y, then we end with x so let's equate let u be equal to line y so that the u would be equal to 1 over y the y n if y equals to y the u so we can substitute that but before then let's look at the limits so we have the upper limit as zero we have a the lower limit as 0 and the upper limit as 1.
07:08
Or learn 0, learn 1 that is called to 0...