Surface area of cap using integrals

In summary, the conversation discusses how to find the surface area of a cap cut from a sphere and a cone. The solution involves using the equation cos(v) = n*.k/|n| and converting to spherical coordinates. The final answer should be 2pi(2-sqrt(2)). The individual is struggling with using the formula in an integral and asks for assistance. A suggestion is made to use spherical coordinates to simplify the problem.
  • #1
lilmul123
40
0

Homework Statement


The question asks,

"Find the surface area of the cap cut from the sphere x^2+y^2+z^2=2 by the cone z = sqrt(x^2+y^2)" The answer should be 2pi(2-sqrt(2))

My main problem is not knowing how to get started.

Homework Equations



With the example given, it seems we need to find cos(v) first using the equation cos(v) = n*.k/|n|.

The Attempt at a Solution



I found the normal line to be 2xi+2yj+2zk. Using the above formula, I eventually reached the conclusion that z/sqrt(r^2+z^2). I don't know how to use this in an integral and it doesn't follow the example our professor gave us either. Can anyone help?
 
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  • #2
Tame this problem by writing:

z = r cos (theta) (here theta is the zenith)

and

r^2 = x^2 + y^2 + z^2.

Work in spherical coordinates. It'll be that much easier.
 

1. What is the formula for finding the surface area of a cap using integrals?

The formula for finding the surface area of a cap using integrals is:S = 2π∫(y√(1+(dy/dx)^2)dx)

2. How do you set up the integral for finding the surface area of a cap?

To set up the integral, you need to first find the limits of integration by determining the points of intersection between the curve and the x-axis. Then, plug in the values into the formula: S = 2π∫(y√(1+(dy/dx)^2)dx).

3. Can you use any type of curve to find the surface area of a cap using integrals?

Yes, you can use any type of curve as long as it is a function of x and the limits of integration can be determined.

4. Is there a simpler way to find the surface area of a cap without using integrals?

Yes, you can use the formula S = 2πrh, where r is the radius of the cap and h is the height of the cap. This formula only works for circular caps.

5. What is the practical application of finding the surface area of a cap using integrals?

Finding the surface area of a cap using integrals is useful in engineering and physics, specifically in calculating the surface area of objects with irregular shapes. It can also be used in calculating the amount of material needed to cover a surface or the amount of paint needed to cover a curved surface.

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