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Surface Area of a Sphere
In this example we will complete the calculation of the area of a surface of rotation. If we’re going to go to the effort to complete the integral, the answer should be a nice one; one we can remember. It turns out that calculating the surface area of a sphere gives us just such an answer. Remember that in an earlier example we computed the length of an infinites- imal segment of a circular arc of radius 1:
ds =
1 − x^2
dx
x 1 x 2
Figure 1: Part of the surface of a sphere.
For this example we’ll let the radius equal a, so that we can see how the surface area depends on the radius. Then:
y =
a^2 − x^2 y′^ = −x √ a^2 − x^2
ds =
x^2 a^2 − x^2
dx
a^2 − x^2 + x^2 a^2 − x^2
dx
a^2 a^2 − x^2
dx
The formula for the surface area indicated in Figure 1 is:
Area =
∫ (^) x 2
x 1
2 πy ds
∫ (^) x 2
x 1
2 π
y ︷√ ︸︸ ︷ a^2 − x^2
ds ︷√ ︸︸ ︷ a^2 a^2 − x^2 dx
∫ (^) x 2
x 1
2 π
a^2 − x^2
a √ a^2 − x^2
dx
∫ (^) x 2
x 1
2 πa dx
= 2 πa(x 2 − x 1 )
Special Cases
When possible, we should test our results by plugging in values to see if our answer is reasonable. Here, if we set x 1 = 0 and x 2 = a we should get the surface area of a hemisphere of radius a:
0 a
Figure 2: Right hemisphere.
2 πa(x 2 − x 1 ) = 2 πa(a − 0) = 2 πa^2
We get the surface area of the whole sphere by letting x 1 = −a and x 2 = a:
2 πa(x 2 − x 1 ) = 2 πa(a − (−a)) = 4 πa^2
Question: Would it be possible to rotate around the y-axis? Answer: Yes. If we rotate around the y axis and integrate with respect to x (calculating the surface area of a vertical slice, as we did here) we’d be adding up little strips of area. If we integrate with respect to y and find the surface area between two vertical positions y 1 and y 2 we’d get exactly the same calculation.
Question: Can you compute surface area using shells? Answer: The short answer is “not quite”. We use the word shell to describe something which has a thickness dx. Shells have volume, integrals which involve shells compute volumes, not surface areas.
