Approximating surface area of hemisphere

In summary, the conversation is about approximating the surface area of a hemisphere using cylinders with varying radii. However, this method is proven to be inaccurate as the total length of the cylinders will always be 2, while the actual surface area is \sqrt{2}. The person is asking for an alternative method to integrate the surface area without using the volume method.
  • #1
serverxeon
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0

Homework Statement



i am trying to apprixmate the surface area of a hemisphere.
i am approximating by cutting the sphere into cylinders of different radius, and using their curved surface area to approximate.

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each cylinder will have a height of r.cos.theta and radius of r.sin.theta.

so the surface area should be =
integral [pi/2 to 0] (2.pi * r.sin.theta * r.cos.theta * d.theta)??

but that gives me -pi r^2 which is wrong...

anyidea where i went wrong?
 
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  • #2
That's not going to work. Those cylnders do NOT approximate the surface area. The problem is exactly the same as if you used short vertical and horizontal segments to approximate the line from (0, 0) to (1, 1). They will always have a total length of 2 while the length of the the line is [itex]\sqrt{2}[/itex].
 
  • #3
sorry i don't get you. can you elaborate?
how else can i integrate something to get the S.A.?
(without going through the volume method)
 

Related to Approximating surface area of hemisphere

1. What is the formula for finding the surface area of a hemisphere?

The formula for finding the surface area of a hemisphere is 2πr², where r is the radius of the hemisphere.

2. How is the surface area of a hemisphere different from a full sphere?

A hemisphere is half of a sphere, so its surface area is half of a full sphere's surface area. The formula for a full sphere is 4πr², which is twice the formula for a hemisphere.

3. Can you explain the concept of "approximating" the surface area of a hemisphere?

Approximating means estimating or finding an approximate value. In the case of finding the surface area of a hemisphere, we may not be able to find the exact value due to the curved shape of the surface. Instead, we use a mathematical formula to estimate the surface area.

4. Why is it important to be able to approximate the surface area of a hemisphere?

Being able to approximate the surface area of a hemisphere is important in many real-world applications, such as calculating the surface area of a dome or the volume of a container with a hemispherical top. It allows us to make practical calculations and designs.

5. Are there any other methods for finding the surface area of a hemisphere besides using the formula?

Yes, there are other methods such as using calculus or integration. However, the formula 2πr² is the most commonly used and efficient method for approximating the surface area of a hemisphere.

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