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### Surface Area of Revolution

Main formula:

Surface Area of Revolution =

Hints and tips:

• To remember this formula, it helps to recall that the square root part comes from the arclength formula (which in turn comes from the Pythagorean Theorem). The 2πf (x) part comes from the circumference of a circle, 2πr, where the radius r is the height of the curve f (x) that is revolving around the x-axis.

• Remember that you must integrate the square root formula above. A common mistake is to integrate the function itself, not the square root formula. Of course, this would give you the area under the curve and not the surface area of revolution.

• A similar mistake is to mix this up with the arclength formula, which looks similar. Be careful which one you are asked for.

• Don’t make the common algebraic mistake of thinking that reduces to a + b! This is extremely wrong, and your teacher will likely be merciless if you do it

• A common technique in problems of this nature is to make a u-substitution for whatever is under the square root sign. Then (hopefully) you can manipulate the expression outside the square root (which comes from f (x)dx) into being the du. However, you might have to do several steps of algebraic manipulation, pulling factors in or out of the square root sign, before this works.

• If the graph is being revolved around the y-axis, simply switch the roles of x and y in the formula above. Be sure to check carefully which one the problem is asking for.

• When it’s feasible, check that your answer makes sense. Unlike area integrals, which can be negative if a curve goes below the x-axis, surface area of revolution should always be positive!

## Surface Area of Revolution

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