constant values ??in the x direction and y direction , while the one constant in the z direction

Even if z eÅittirsÄ±f horizontal plane, coordinate plane x minus top

y plus z variables such as A surface definition.

So we d.times.d this function y We will do our integral is over.

But all of them here, of course, where x y z

which is linear for What if we could of.

But if the variable z zero where z is the natural choice as the right

here we find z zero four minus x plus y to the integral.

This will do on our first integral.

Then y or x on the matter, the will do.

Functions can be allocated to variables.

But with the constant value limits is not defined,

that all previous limits components, such as taking

taking the integral of three single-storey bi We can become multiplied.

Let's do this.

The first step, as you can see here We will do our first integral on.

Our function was x squared y z.

here x and y will be affected.

We have only to him.

z z integral square divided by two.

You're writing here.

This z is zero and four negative in the range of x plus y.

Therefore, single storey integral gone.

Just had a two-storey integral.

I'm bringing this y here.

When this function calculator,

z is equal to zero following a course contribution will not hit above y.

The above frame is coming.

Here are a divided by two.

We got out at one-half.

Now we need to do this integral.

This two-storey integral now you know.

In this two-tiered integrally y boundaries of a merger between minus.

Then it will do the integration.

This merger is a minus here Because of little

To simplify calculations I had to choose such.

y is one function decrease.

Therefore, separate them you do not have to account.

Here our remaining term was: Four minus x plus y squared.

X, Y are each independently of terms We believe it to separate, its square,

plus y squared, multiplied by that year, plus four minus two times x times y.

But in a more y'yl multiplied.

y squared.

See terms here y are the only forces.

Therefore, the only forces y y suddenly drops to minus entrgral,

All that remains is the integral term remains.

This term is an integral over the years means that x is hard to say.

Where b x is the square outside.

Here are four minus x.

Two here.

It took one-half of the two.

y the y integral of the square cube divided by three.

And that a negative value will account for.

y is equal to minus one and two

time will change sign two times lower for the future.

Two divided by three is going on here.

Wherein Y is given instead where x is the square for.

I have four.

There are minus x.

The four x squared minus x cube is happening.

The reset value of x.

That the integration of this single-storey, I would easily account.

Two-thirds here.

Four x cube minus x divided by three to four divided by four.

x is zero and the values ??of a.

Results so simply involved.

This is the third example of a homework as is given to you.

There are still zero on x.

on y zero.

zero on z.

But I take the plane.

This plane is equal to x than z equals one, the plane y equals one.

This plane has a volume formed.

We want to have this volume.

In addition, this volume is with v, z, hes We want to calculate the integral.

It has a meaning.

If you divide the integral take it to v

that the weight on the z axis You can find the location of the center.

If you do not want to deal with it you an example of integration.

Now how do we do it?

Now see here a bit more complicated than the previous.

Because z, z on the start of integrals.

Scratch that you find here is equal to z integration leading to a minus x minus y.

But then the remaining integral over x and y.

But this plane, x and y axis z is zero at x

cuts along the line y equals a plus.

To coordinate plane also is There are currently integral on the triangle.

If we write this more evident this figure

We need to do the integral over.

That accounts integrally You are welcome to.

Here we make it easy as shown in these results.

It answers You can control.

The next sample spherical coordinate problem.

The volume of the sphere.

We know all of us.

Four pi squared divided by three.

If a radius.

We want to see how it found.

In this cylindrical coordinates, We want to make in spherical coordinates.

Moreover globe, is a rotating objects.

You receive a circle.

You can returns.

Turns out the surface of the sphere.

And when we do it for We want to find the volume of the sphere.

We want to make this example.

Now I want to take a break.

After which the gene We will continue to sample.

Ball and roller considerably coordinates example will do.

The reason is clear.

Because these shapes in nature and technology very frequently encountered volumes.

Goodbye.