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8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

Part of Applications of Integration in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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This topic is BC only. The arc length of a smooth curve y=f(x) on [a,b] is
L=∫ab1+[f′(x)]2 dxL=\int_a^b \sqrt{1+[f'(x)]^2}\,dx
This formula comes from approximating the curve with many tiny line segments and then taking a limit. If the curve represents the path of a moving particle in a plane, then arc length gives the total distance along the path. In motion contexts, this is closely connected to distance traveled, especially when the motion follows a curve rather than a straight line.

The key idea is that arc length measures the actual path length, not just horizontal or vertical change. The derivative f'(x) enters because it determines the slope of each tiny segment, and the square root expression comes from the Pythagorean Theorem. You should recognize that this formula is more complex than most earlier integral setups because the integrand itself depends on the derivative. In some cases, the integral evaluates cleanly; in others, it may be left in exact integral form.

In this unit

13 lessons

  1. 8.1Finding the Average Value of a Function on an IntervalPreview lesson
  2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using IntegralsPreview lesson
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied ContextsPreview lesson
  4. 8.4Finding the Area Between Curves Expressed as Functions of xPreview lesson
  5. 8.5Finding the Area Between Curves Expressed as Functions of yPreview lesson
  6. 8.6Finding the Area Between Curves That Intersect at More Than Two PointsPreview lesson
  7. 8.7Volumes with Cross Sections: Squares and RectanglesPreview lesson
  8. 8.8Volumes with Cross Sections: Triangles and SemicirclesPreview lesson
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-AxisPreview lesson
  10. 8.10Volume with Disc Method: Revolving Around Other AxesPreview lesson
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-AxisPreview lesson
  12. 8.12Volume with Washer Method: Revolving Around Other AxesPreview lesson
  13. 8.13The Arc Length of a Smooth, Planar Curve and Distance TraveledThis lesson

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