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Compute the arc length of the curve
between the endpoints and .
First we compute the derivative:
Next, we write the arc length element:
The key to integrating is to first fully simplify the quantity inside the square
When the middle term in a binomial (i.e., FOIL) expansion is adding
one always gives a perfect square. In this case, the algebraic fact is
The general phenomenon is that what was formerly the square of a
difference—in this case, —becomes the square of the corresponding