Find the surface area of the part of the sphere x2+y2+z2=81 that lies above the cone z=√(x2+y2)
Find the surface area of the part of the sphere x^2+y^2+z^2=81 that lies above the cone z=√(x^2+y^2)
the cone intersects the sphere where
x^2+y^2+(x^2+y^2) = 81
x^2+y^2 = 81/2
that is a circle of radius 9/√2 in the plane z=9/√2
That means the spherical cap has a height of 9(1-1/√2). There are many sites which can give you the formula for that.
As for the integral, it looks like you want
4∫[0,9/√2] ∫[0,√(81/2-x^2) dS
Check for the surface area element.
You could also use spherical coordinates which would look less complicated.
How do i set up the integral?
To find the surface area of the part of the sphere that lies above the cone, we need to consider the intersection of these two surfaces.
First, let's find the intersection curve between the sphere and the cone. We can do this by setting their equations equal to each other:
x^2 + y^2 + z^2 = 81
z = √(x^2 + y^2)
We can substitute the expression for z from the second equation into the first equation:
x^2 + y^2 + (√(x^2 + y^2))^2 = 81
x^2 + y^2 + x^2 + y^2 = 81
2x^2 + 2y^2 = 81
x^2 + y^2 = 40.5
This is the equation of a circle in the xy-plane with a radius of √40.5.
Now, to find the surface area above the cone, we need to integrate the surface area element over this region. The surface area element is given by:
dS = √(1 + (dz/dx)^2 + (dz/dy)^2) dA
Where dA is the area element in the xy-plane.
We can calculate dz/dx and dz/dy by taking the partial derivatives of z with respect to x and y, respectively:
dz/dx = x/√(x^2 + y^2)
dz/dy = y/√(x^2 + y^2)
Now, we can evaluate the integral of the surface area element over the region x^2 + y^2 = 40.5:
S = ∫∫√(1 + (x/√(x^2 + y^2))^2 + (y/√(x^2 + y^2))^2) dA
We can convert this integral to polar coordinates by substituting x = r*cosθ and y = r*sinθ:
S = ∫∫√(1 + (r*cosθ/√(r^2))^2 + (r*sinθ/√(r^2))^2) r dr dθ
Simplifying the expression inside the square root and evaluating the integral over the region defined by 0 ≤ r ≤ √40.5 and 0 ≤ θ ≤ 2π will give the surface area of the part of the sphere that lies above the cone.
A more general solution is found below; you can plug in your numbers.
http://www.freemathhelp.com/forum/threads/70982-surface-integrals-spherical-cap