Cogito
AP Calculus BC · Chapter 7 · Lesson 2
Arc Length and Surface Area
Measuring a curve rather than the area under it.
12 problems · about 21 minutes · CHA-6.A
What this lesson teaches
The student sets up integrals for arc length in rectangular and parametric form.
- Arc length integrates √(1 + (dy/dx)²) in rectangular form.
- In parametric form it integrates √((dx/dt)² + (dy/dt)²).
- Both are Pythagoras, and the marks are usually for the setup.
Warm Up
Straightforward practice. Get the method working first.
5 problemsdx/dt = 9 and dy/dt = 12. What is the speed?
Answer 15
Why 15.
Where does the arc length formula come from?
Answer Pythagoras applied to an infinitesimal step along the curve.
Why Pythagoras on a tiny triangle.
dx/dt = 6, dy/dt = 8. Speed?
Answer 10
Why √100.
dx/dt = 5, dy/dt = 12. Speed?
Answer 13
Why √169.
In the rectangular arc length formula, what constant sits under the root?
Answer 1
Why The horizontal step is dx.
Build It Up
The same ideas with more to keep track of.
3 problemsy = x², so dy/dx = 2x. What is dy/dx at x = 5?
Answer 10
Why 2 × 5.
y = 3x. What is dy/dx?
Answer 3
Why A constant slope.
Do arc length integrals usually have elementary antiderivatives? 1 yes, 0 no.
Answer 0
Why They rarely do.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each form with its integrand.
Answer Rectangular arc length → Root of one plus dy by dx squared; Parametric arc length → Root of the two rates squared and added; Both formulas → Are Pythagoras on an infinitesimal triangle
Why The 1 appears when the horizontal step is dx itself.
dx/dt = 8, dy/dt = 15. Speed?
Answer 17
Why √289.
The Derivative: y = x². What is dy/dx at x = 3?
Answer 6
Why 6.
The Step: dx/dt = 3 and dy/dt = 4. What is √((dx/dt)² + (dy/dt)²)?
Answer 5
Why 5.