Cogito
Multivariable Calculus · Chapter 2 · Lesson 3
Arc Length and Curvature
How long the path is, and how sharply it bends.
12 problems · about 23 minutes · N-VM.A.1
What this lesson teaches
The student sets up arc length integrals and interprets curvature.
- Arc length is the integral of speed, because distance is speed accumulated over time.
- Constant speed turns the integral into a multiplication.
- Curvature is how fast direction turns; a circle of radius R has curvature 1/R.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA circle of radius 25. What is its curvature, as a decimal?
Answer 0.04
Why 0.04.
What is being integrated in an arc length integral?
Answer The speed, which is the magnitude of the velocity.
Why Speed, integrated over the time interval.
Speed is constant at 9 from t = 0 to t = 4. What is the arc length?
Answer 36
Why Speed times time.
A circle of radius 4. What is its curvature, as a decimal?
Answer 0.25
Why 1 over 4.
A straight line. What is its curvature?
Answer 0
Why The direction never turns.
Build It Up
The same ideas with more to keep track of.
3 problemsCurvature is 0.2. What is the radius of curvature?
Answer 5
Why Take the reciprocal.
r(t) = ⟨5t, 12t, 0⟩ from t = 0 to t = 3. What is the arc length?
Answer 39
Why Speed 13 for 3 units of time.
Which bends more sharply, a circle of radius 2 or one of radius 10? Enter the radius of the sharper one.
Answer 2
Why Smaller radius, larger curvature.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder these paths from least curved to most curved at their tightest point.
Answer 1. A straight road 2. A gentle motorway curve of radius 500 m 3. A roundabout of radius 20 m 4. A hairpin bend of radius 5 m
Why Curvature is one over the radius.
The speed of a particle is 0 for a whole interval. What is the arc length over that interval?
Answer 0
Why Nothing moved.
The Track: A cart moves at constant speed 6 for 7 seconds. How far does it travel?
Answer 42
Why 42.
The Bend: A circular bend of radius 20 metres. What is its curvature, as a decimal?
Answer 0.05
Why 0.05.