Arc Length (AB/BC)

AP Calculus · Axiom Academy

Measuring the length of curves using integration Step 1: From Line Segments to Integrals How do you measure the length of a curve? The idea is beautifully simple: approximate the curve with many short line segments , then take the limit as the segments get infinitely small. Each tiny segment has horizontal width and vertical rise . By the Pythagorean theorem, the segment length is: Factor out from under the radical: As , the ratio becomes the derivative , and the sum of all segments becomes a definite integral. Step 2: The Arc Length Formula Taking the limit of our Riemann sum gives us the exact arc length: For curves given parametrically as : Step 3: Setting Up an Arc Length Integral Step 5: When Exact Evaluation Isn't Possible Unlike many integrals you have seen, arc length integrals are frequently impossible to evaluate in closed form . Even simple-looking curves like produce integrands with no elementary antiderivative. On the AP exam, you have two strategies: Given a table of values for , approximate using left, right, midpoint, or trapezoidal sums. Set up the integral correctly, then state you would evaluate it numerically. On the BC exam's calculator section, use your graphing calculator's numerical integration feature.

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