Labware - MA35 Multivariable Calculus - Single Variable Calculus
 MA35 Labs 1 » Single Variable Calculus Contents1.4 Integration 1.4.2 Riemann Integral 1.4.4 Arc Length for Function Graphs 1.4.5 Change of Variables 1.4.7 Center of Mass Search

Arc Length for Function Graphs

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We can approximate the arc length of the function curve for a function f(x) by breaking the curve into segments of tangent lines.

A segment that starts at the point (x, f(x)) and ends a distance dx to the right will end at the point (x + dx, f(x + dx)) = (x + dx, f(x) + f '(x)dx). This means the length of the segment is √(dx2+(f '(x)dx)2). This expression simplifies to √(1 + (f '(x))2)dx.

Integrate with respect to x from a to b, resulting in the following formula for arc length s:

s = ∫ab√(1 + (f '(x))2)dx

Demos

 Arc Length Here is a visualization of approximation of arc length by segments of tangent lines. Increase the resolution to get a better approximation. Try changing f(x) as well.

Exercises

• 1. Use the demo to relate (using <, >, =) the arc lengths from -1 to 1 of the functions f(x) = 0, f(x) = x, f(x) = x2, and f(x) = x3. Explain from the formula for arc length why the comparison turns out this way.
• 2. How would one go about using the formula for arc length to find the arc length of a function which is not continuously differentiable (e.g. f(x) = |x|)?