How to Find the Volume Bounded by a Cylinder and Planes in the First Octant?

[mathrocks]

I'm kinda lost on this problem

"Find the volume of the give solid:
Bounded by the cylinder y^2+z^2=4 and the planes x=2y, x=0, z=0 in the first octant."

Any help would be appreciated.
Thanks!

 

[theFuture]

First thing that helps is to draw a picture. Then, you'll see that it's helpful here to work in cylindrical coordinates. Are you stuck on the limits of integration or where?

 

[doriang101]

Volume of region

In this case it doesn't seem like you need to use double integrals. you have y^2+z^2 = 4 and y = x/2 for one of the planes. If you draw the xy plane you will just have the line y=x/2. If you visualize any cross section of the cylinder made by placing a plane perpindicular to the xy and perpindicular to the xz plane, that the height of the cylinder at that point (the value of z) will be constant along the plane. So what I would do is, Integrate along the y-axis by breaking down the cylinder into shells.

so you get V = Integral(2*y*sqrt(4-y^2)dy, 0, 2). Which is solved by a simple substitution.

 

[mathrocks]

First thing that helps is to draw a picture. Then, you'll see that it's helpful here to work in cylindrical coordinates. Are you stuck on the limits of integration or where?

That's the thing, I don't really understand how to do it with cylindrical coordinates.

 

[mathrocks]

In this case it doesn't seem like you need to use double integrals. you have y^2+z^2 = 4 and y = x/2 for one of the planes. If you draw the xy plane you will just have the line y=x/2. If you visualize any cross section of the cylinder made by placing a plane perpindicular to the xy and perpindicular to the xz plane, that the height of the cylinder at that point (the value of z) will be constant along the plane. So what I would do is, Integrate along the y-axis by breaking down the cylinder into shells.

so you get V = Integral(2*y*sqrt(4-y^2)dy, 0, 2). Which is solved by a simple substitution.

I think my teacher wants me to use double integrals...

 

[theFuture]

ok, well first you should set it up so V=int(function) dxdydz. Then we'll change the function into cylindrical coordinates. To do this, remember x=rcostheta and y=rsintheta and z=z. You also need the Jacobian here.

 

[mathrocks]

ok, well first you should set it up so V=int(function) dxdydz. Then we'll change the function into cylindrical coordinates. To do this, remember x=rcostheta and y=rsintheta and z=z. You also need the Jacobian here.

Why is there a dz in there, shouldn't it only be a double integral, since it's in that particular section?

 

[arildno]

Note that we have:
$$0\leq{x}\leq{2y}$$
Also, we have: $$0\leq{z}\leq\sqrt{4-y^{2}}$$
While $$0\leq{y}\leq{2}$$
Hence, we have the double integral V:
$$V=\int_{0}^{2}\int_{0}^{\sqrt{4-y^{2}}}2ydzdy$$

 

[Theelectricchild]

Always recall that when finding volumes, these double integrals are simply "simplified" triple integrals.

The definition of volume is just the triple integral of a volume element dV.

Therefore

$$V=\int\int\int_E dV$$

when the first integral is carried out, you are left with the integrand that arildno pointed out above this post.

 

[Theelectricchild]

A dumbed down version of the definition I should point out is:

Volume is the triple integral of 1.