Problems Plus 4: Curvature from Parametric Equations that involve Integrals

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In this video, I show how to obtain the curvature of parametric equations that involve integrals, by first utilizing the fundamental theorem of calculus to obtain the derivative of the position vector from the integrands of the integrals. Then we can follow the usual steps to obtain the curvature, which is defined as the magnitude of the unit tangent vector divided by the magnitude of the derivative of the position vector.

Timestamps

Problem 4: Curvature from Parametric Equations – 0:00
Solution: Recall the definition of curvature as the magnitude of the derivative of the unit tangent vector in terms of arc length – 0:22
Derivative of position vector from the Fundamental Theorem of Calculus (FTC) – 1:57
Unit Tangent Vector becomes the first derivative of position vector – 5:00
Magnitude of derivative of position vector equals π|t| – 8:30
Curvature is also equal to π|t| – 10:30
Recall Parts 1 and 2 of the FTC – 11:11

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Full written solution: Vector Functions: Problems Plus
More math: mes.fm/math

Solution to Problem 4

Full written solution — mirrored from Vector Functions: Problems Plus, which also covers Problems 1–3 and 5–7.

Find the curvature of the curve with parametric equations:

Solution:

Recall the equation of curvature and the unit tangent vector from my earlier video.

By the Fundamental Theorem of Calculus:

Recall my earlier videos on Part 1 and Part 2 of the Fundamental Theorem of Calculus.

Originally posted on the Hive blockchain →

Text and image retrieved from the Hive blockchain on September 21, 2026.