00:01
Hi there, so for this problem we are asked to evaluate the below integral where e is the solid bounded by the cylinder and that cylinder is y is equal to the square root of x and the planes y equals to zero, z equals to zero and then x plus z is equal to pi divided by two and that the integral in question for this is the triple integral of y times the cosine of x plus z and then this integrated over x, y and z.
00:46
The first thing that we need to do is to find, is to set up the bounds of integration for each variable, okay? so firstly let's consider the bounds for z.
01:01
Since the solid e is bounded above by the plane that is x plus z that is equal to pi divided by two, we know that z is bounded, we know that z is bounded above by z that is equal to pi divided by two minus x, okay? and meanwhile the solid is bound below by the plane z equals to zero, right? now let's do for y.
01:35
Now we know that the solid e is bounded by the cylinder that is y equal to the square root of x and so y is bounded above by this function and since the solid is also bounded below by the function y equals to zero, we know that y is then bounded below by zero, right? and for x, for each value of y and z the solid e is bounded by the curve that we know is the square root of x which implies that x is bounded below by x equals to y square, okay? meanwhile the solid is only bounded by the plane that we know x plus y, let's see it equals to pi divided by two which implies that x is bounded above by what we can solve in there and to be just simply pi divided by two minus z.
02:35
So using these bounds we can just now write the following integral.
02:41
So we will have the integral and we go from zero to pi divided by two.
02:47
In here we start by the, by the, okay so in here we start first with the bounding for z.
03:06
Let's go with the next bounding and here we go from zero to the square root of x or y and finally we go from zero to pi divided by two minus x.
03:21
Oh sorry, so in this case this first one in here will be for z, okay? so this, and the function that is y times the cosine of x plus z, this integrated over x, y and z.
03:56
Okay, well in here, let me fix that in here.
04:08
So let's first do the integral over z, okay? so that will give us that this is zero from, from zero to pi divided by two, then from zero to the square root of x.
04:23
Now since in here nothing depends on z except for the, the cosine function, then we will have that this will be just simply, well the derivative, well we will have y in here...