Surface Area of Sphere above xy-Plane & in Cylinder

In summary, the problem is to find the surface area of the portion of the sphere x^2 + y^2 + z^2 = a^2 that lies above the xy-plane and within the cylinder x^2 + y^2 = b^2, where 0 \leq b \leq a. To approach this problem, it is recommended to draw a picture and consider the case when b = a to check the integration.
  • #1
mathwurkz
41
0
Hi! I don't know how to approach this problem. I need a little bit of help please. Here is the problem:
Find the surface area of that portion of the sphere [tex]x^2 +y^2 + z^2 =a^2 [/tex] that is above the xy-plane and within the cylinder [tex] x^2 + y^2 = b^2, 0 \leq b \leq a[/tex]
 
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  • #2
Give us some input on the integral youre thinking about..
 
  • #3
mathwurkz said:
Hi! I don't know how to approach this problem. I need a little bit of help please. Here is the problem:
Find the surface area of that portion of the sphere [tex]x^2 +y^2 + z^2 =a^2 [/tex] that is above the xy-plane and within the cylinder [tex] x^2 + y^2 = b^2, 0 \leq b \leq a[/tex]

Two things Mathwurkz:

1. That's not a surface integral but rather a calculation to determine the surface area. Surface integrals are different. Check them out if you wish.

2. Draw a picture: Ideally, draw one in 3-D using Mathematica or other software. But even a cross section would be helpful. Once you have an accurate picture in mind, it's easier to construct the integral for the surface area.

Edit: One more thing Mathwurkz:

What happens when a=b? Then the problem is easy right? Anyway, that's a good way to check your integration: If it works for this simple case which is known by inspection, then that gives some confidence it's correct when b is less than a.
 
Last edited:

1. What is the formula for calculating the surface area of a sphere above the xy-plane?

The formula for calculating the surface area of a sphere above the xy-plane is 4πr2, where r is the radius of the sphere.

2. How do you find the surface area of a cylinder?

The formula for finding the surface area of a cylinder is 2πr2 + 2πrh, where r is the radius of the base and h is the height of the cylinder.

3. Can the surface area of a sphere above the xy-plane and in a cylinder be calculated separately?

Yes, the surface area of a sphere above the xy-plane and in a cylinder can be calculated separately using the formulas mentioned above.

4. Is the surface area of a sphere above the xy-plane and in a cylinder the same?

No, the surface area of a sphere above the xy-plane and in a cylinder are not the same. The surface area of a sphere above the xy-plane is only a part of the total surface area of the sphere, while the surface area in a cylinder includes both the curved surface and the circular bases.

5. How does the radius of the sphere and cylinder affect their surface area?

The surface area of a sphere and cylinder are directly proportional to the square of their radius. This means that if the radius is doubled, the surface area will be four times larger.

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