Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

bodensee9 · Jan 31, 2007

Homework Statement

I am wondering if someone could help me with the following? I am supposed to find the area of the finite part of the paraboloid y = x^2+z^2 that's cut off by the plane y = 25. Now, wouldn't this be the same as the paraboloid z = x^2+y^2 that's cut off by the plane z = 25?

So, if that's right, then the surface area is ∫∫√((dz/dx)^2+(dz/dy)^2+1).
This seems easier to do in polar coordinates, so we basically have the following:

∫∫(1+4r^2)? And, r would be from 0 to 5, and 0≤θ≤2*pi?

Am I doing something wrong here? Or, how do I "project" the surface unto the x-z plane? Thanks!

Dick · Jan 31, 2007

Why do you think you are doing something wrong? Well done.

Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

Homework Statement

1) What is the formula for finding the surface area of a paraboloid cut by a plane?

2) How do you determine the boundaries for the integral when finding the surface area of a paraboloid cut by a plane?

3) Can you use the same formula to find the surface area of any paraboloid cut by a plane?

4) How do you interpret the result of the surface area calculation for a paraboloid cut by a plane?

5) Are there any practical applications for finding the surface area of a paraboloid cut by a plane?

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