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Cogito

Calculus · Chapter 7 · Lesson 1

Practice Worksheet

Area Under a Curve

Accumulation, before any antiderivative.

10 problems · about 22 minutes · LIM-5.A, LIM-5.B

NameDate

Figure — use these to answer the problems

Move the two ends and watch the accumulated area change.
-8-8-6-6-4-4-2-222446688
y = 1x + 1
  • Point(0, 1)
  • Second point(4, 5)
  • Slope between them1
Slope1
Intercept1
From x0
To x4
Figure 1

Warm-Up Review

From earlier lessons. Loosen up before the new work.

2 problems

  1. From “Optimisation”

    Two numbers add to 16. What is their greatest product?

    Answer

  2. From “The Second Derivative and Concavity”

    f′(4) = 0 and f″(4) = −3. Max or min? 1 max, 2 min.

    Answer

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. Speed rises 0 to 12 m/s over 10 s. Distance?

    Answer

  2. What is a definite integral?

    • The limit of sums of ever-thinner rectangles.
    • The opposite of a derivative.

  3. A car at 12 m/s for 5 s. Distance?

    Answer

  4. Speed rises 0 to 8 m/s over 4 s. Distance?

    Answer

Build It Up

The same ideas with more to keep track of.

2 problems

  1. A car at 25 m/s for 8 s. Distance?

    Answer

  2. Under a speed-time graph, area represents what?

    • Distance.
    • Acceleration.

Stretch Yourself

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

2 problems

  1. The Steady Speed

    A car at 15 m/s for 4 seconds. What distance?

    Answerm

  2. The Triangle

    Speed rises steadily from 0 to 10 m/s over 6 seconds. What distance?

    Answerm