Find the volume of the region bounded by parabolic cylinder and planes

[mharten1]

Homework Statement

Find the volume of the solid bounded by the parabolic cylinder y = x^2 and the planes z = 3-y and z = 0

Homework Equations

The Attempt at a Solution

Obviously, a triple integral must be used in the situation. Our professor never explained how to find the limits of integration, this is the part that is confusing me. I believe that the limits for z are 3-y and 0. However, I have no idea how to get the limits of x and y. Any help is greatly appreciated.

 

[tiny-tim]

hi mharten1!

… the limits for z are 3-y and 0. However, I have no idea how to get the limits of x and y. Any help is greatly appreciated.

no, your first limits (for z, say) will just be two numbers (no y)

your next limits (for y, say) will depend on z

your third limits will depend on y and z

standard method: cut the region into horizontal slices between heights z and z+dz …

what is the 2D shape of each slice? ​

 

[Dick]

Basically you want to draw a sketch of the solid you are trying to find the volume of. Can you visualize it? z=0 is the x-y plane. Where does the plane z=y-3 intersect the x-y plane?

 

[mharten1]

Basically you want to draw a sketch of the solid you are trying to find the volume of. Can you visualize it? z=0 is the x-y plane. Where does the plane z=y-3 intersect the x-y plane?

Doesn't it intersect it when z=0? So at y=3?

 

[Dick]

Doesn't it intersect it when z=0? So at y=3?

Sure. So what region do you want to integrate z over in the x-y plane?

 

[mharten1]

Sure. So what region do you want to integrate z over in the x-y plane?

The region from 0 to 3? If that's not right, I guess I'm not visualizing this in the right way. I think I'll graph out the region so I can better see it.

 

[Dick]

The region from 0 to 3? If that's not right, I guess I'm not visualizing this in the right way. I think I'll graph out the region so I can better see it.

I think you should graph it out. You should anyway. "The region from 0 to 3?" isn't a very accurate description.