# Converting Area to Volume: A Spherical Coordinate Approach

• brioches
In summary, the conversation discusses finding the general expression for an infinitesimal area element in spherical coordinates, using n as the outward-pointing normal vector. The speaker suggests using a triple integral, similar to the one used for finding the volume of a sphere, to integrate the area element. Another speaker suggests looking up the area element or working it out from first principles.
brioches

## Homework Statement

A hemispherical surface of radius b = 61 m is fixed in a uniform electric field of magnitude E0 = 3 V/m as shown in the figure. The x-axis points out of the screen.

Enter the general expression for an infinitesimal area element dA in spherical coordinates (r, θ, φ) using n as your outward-pointing normal vector. In these coordinates θ is the polar angle (from the z-axis) and φ is the azimuthal angle (from the x-axis in the x-y plane).

φ(flux) = ∫ E dA

## The Attempt at a Solution

I understand that we're supposed to be looking for a small piece of area, but I don't know what makes up that area. We had a triple integral last semester of ∫∫∫r2dr sin(θ)dθ dφ. Is that related? We used that to integrate the volume of a sphere. Is there a similar process one can use for area?

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I guess you can either look up the area element in spherical coordinates or work it out for yourself from first principles.

brioches said:
∫∫∫r2dr sin(θ)dθ dφ. Is that related?
Very much so. Suppose you had an area element dA for a shell radius r within the sphere. How would you turn it into a volume element? Compare that with the integrand above.

## 1. How do you calculate the surface area of a hemisphere?

In order to find the surface area of a hemisphere, you must first find the radius of the hemisphere. Then, you can use the formula A = 2πr^2 to calculate the surface area. Make sure to use the radius of the hemisphere, not the diameter.

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

The formula for finding the surface area of a hemisphere is A = 2πr^2, where A represents the surface area and r represents the radius of the hemisphere.

## 3. Can you explain the concept of "dA" in relation to a hemisphere?

dA stands for differential surface area and it represents an infinitesimal change in the surface area of a hemisphere. It is used in calculus to calculate the total surface area of a hemisphere by breaking it down into smaller, more manageable sections.

## 4. What units are used to measure surface area?

Surface area is typically measured in square units, such as square centimeters (cm^2) or square inches (in^2).

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

Yes, there are multiple methods for finding the surface area of a hemisphere. Some other methods include using the circumference of the base and the height of the hemisphere, or using the volume of the hemisphere and its radius. However, the most commonly used method is the formula A = 2πr^2.

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