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How to Compute Arc Length with Integration

The formula for computing arc length with integration derives from the basic principles of calculus and the Pythagorean theorem. If you imagine zooming in on a very small portion of the arc, you see that the curve approximates the hypotenuse of a right triangle having sides dx, the change in x; and dy, the change in y. Applying the Pythagorean theorem gives the hypotenuse as sqrt((dx)^2 + (dy)^2), and integrating along the arc gives the arc length.

Instructions

- 1
  Write down the formula S(a,b) sqrt(1 + (dy / dx)^2) dx, where S(a,b) represents an integral from a to b. Recall that dy / dx is another way to state the derivative of your function, f(x).
- 2
  Plug the derivative of your function, f(x), into the formula in place of dy / dx. Change the limits of integration, a and b, to the starting and ending x-values of the arc, respectively.
- 3
  Perform the operations in the formula. It may help you integrate if you rewrite the square root as a one-half power.

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