00:01
Hi, in the given problem we have a plane whose equation is z is equal to x plus y and that is the top plane and the plane at the bottom has an equation of z is equal to zero whereas the curve on the back has a plane has an equation of x is equal to y square and the near surface has a equation of y is equal to x square.
00:32
So we have to set up the integral as x is between 1 and 0, 0 or 1.
00:43
So if x is in the right place then we can say that x square is less equal to y and is further less equal to square root of x and z value is less equal to 5x plus y and it's greater equal to 0.
01:06
So the triple integral set up would be triple integral of 3xy will be integrating with respect to z and then with respect to y and then with respect to x.
01:23
So with respect to x it is changing from 0 to 1.
01:26
This is from x square to root over x and z is from 0 to 5x plus y.
01:38
Now let's solve this integral and so here we will be solving this the inner most part from 0 to 1.
01:47
This is from x square to root over x.
01:52
This would be 3xy and z.
01:59
Now z varies from 0 to 5x plus y and this whole result is 0 integrated with respect to y and then with respect to x.
02:10
So on integrating with respect to z this is what we have got with the limit 0 to 5x plus y.
02:17
So that would be 0 to 1 and x square to root over x.
02:24
This is 3xy times 5x plus y dy dx.
02:37
Now let's integrate with respect to y.
02:39
So we can write this as 0 to 1 x square to root over x.
02:48
So this would be 15x square y plus 3xy square and then integrate with respect to y and then with respect to x.
03:06
Next let's integrate with respect to y.
03:10
So that would be from 0 to 1.
03:13
This would be 15x square y square over 2 plus 3xy cube over 3 and the limit will be from x square to root over x and this result will be integrated with respect to x.
03:30
So this would be 0 to 1.
03:33
Now 15x square root x will be x.
03:37
So root x whole square is x...