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Cogito

AP Calculus BC · Chapter 7 · Lesson 2

Practice Worksheet

Arc Length and Surface Area

Measuring a curve rather than the area under it.

12 problems · about 21 minutes · CHA-6.A

NameDate

Figure — use these to answer the problems

Each tiny piece of a curve is the hypotenuse of a right triangle with legs dx and dy.
adjacent = 3opposite = 4hyp = 5
Horizontal step3
Vertical step4
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dx/dt = 9 and dy/dt = 12. What is the speed?

    Answer

  2. Where does the arc length formula come from?

    • Pythagoras applied to an infinitesimal step along the curve.
    • It is an unrelated definition.

  3. dx/dt = 6, dy/dt = 8. Speed?

    Answer

  4. dx/dt = 5, dy/dt = 12. Speed?

    Answer

  5. In the rectangular arc length formula, what constant sits under the root?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. y = x², so dy/dx = 2x. What is dy/dx at x = 5?

    Answer

  2. y = 3x. What is dy/dx?

    Answer

  3. Do arc length integrals usually have elementary antiderivatives? 1 yes, 0 no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each form with its integrand.

    Draw a line from each item on the left to its match on the right.

    • Rectangular arc length
    • Parametric arc length
    • Both formulas
    • Root of one plus dy by dx squared
    • Root of the two rates squared and added
    • Are Pythagoras on an infinitesimal triangle

  2. dx/dt = 8, dy/dt = 15. Speed?

    Answer

  3. The Derivative

    y = x². What is dy/dx at x = 3?

    Answer

  4. The Step

    dx/dt = 3 and dy/dt = 4. What is √((dx/dt)² + (dy/dt)²)?

    Answer