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Surface Area (Calculus)

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Calculus Problem Solving > Surface Area Calculus

The surface area generated by a volume of revolution (for example, the surface of a sphere) can be approximated with calculus. One technique is to find a curve, that when rotated around an axis, creates the same shape as the surface area we are trying to find. We can then find the area of the surface of revolution with integration.

Surface Area of Revolution Formulas

If you have a non-negative smooth function on the closed interval [a, b], the surface area generated by revolving the curve around the x-axis between a and b is given by the following integral:
surface area calculus

If you are revolving the curve around the y-axis, the integral becomes:
surface integral formula rotating around y axis

Example: What is the surface area created when y = x3 is revolved around the x-axis on [0, 2]?

surface area calculus graph

Surface of revolution created by rotating y = x3 (white curve in center) around the x-axis.


Step 1: Plug the given information into the formula. You have three parts to swap out: the bounds of integration, the given function, and the derivative of the function. For this example, the derivative of x3, which is 3x2 is found with the power rule:
surface area calculus formula

Step 2: Integrate. This integral uses u-substitution and the product rule. The solution is 203 (Click here for a PDF of Solution Steps (Symbolab)).

Reasoning Behind the Formula for Surface Area

It may seem strange that you can integrate a three-dimensional shape, like a cone. Up until this point in calculus, you may have been integrating a simple area under a curve. However, nothing different is happening here. What you’re actually doing is flattening the cone, then integrating.


The underlying formula for finding the surface area begins with a right circular cone.

right circular cone

Right circular cone with bas r and slant height l.



The formula for the flattened surface area (S) of this type of cone is [1]:
S = π r l.

To be able to integrate (find the area) with this formula, you slice the cone along the slant edge, (l in the above image) and flatten it.

flat right circular cone

A flattened right circuar cone, cut along slant edge l.



The key idea is that if we rotate a short, straight line segment, the basic shape is a right circular cone or a truncated cone (a cone with the top snipped off). Even if the slant edge of the cone if a curve, integrating gives a very good approximation for surface area.

References

3D Graph created with Desmos.com.
[1] Protter, M. & Protter, P. (1988). Calculus with Analytic Geometry. Jones & Bartlett.

CITE THIS AS:
Stephanie Glen. "Surface Area (Calculus)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/surface-area-calculus/
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