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Cogito

AP Calculus AB · Chapter 6 · Lesson 2

Practice Worksheet

The Fundamental Theorem of Calculus

The two operations undo each other.

12 problems · about 22 minutes · FUN-5.A, FUN-6.A

NameDate

Figure — use these to answer the problems

Move the two edges and watch the shaded area. The evaluation part computes it without adding rectangles.
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y = 1x² + 0x + 0
  • Point(0, 0)
  • Second point(2, 4)
  • Slope between them2
a1
Lower limit0
Upper limit2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. g(x) is the integral of t² from 1 to x. What is g′(4)?

    Answer

  2. What does the Fundamental Theorem establish?

    • Differentiation and integration are inverse operations.
    • That area can be approximated by rectangles.

  3. Integral of 2x from 0 to 4, antiderivative x². Value?

    Answer

  4. Integral of 3x² from 0 to 2, antiderivative x³. Value?

    Answer

  5. g(x) is the integral of t³ from 0 to x. What is g′(2)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. g(x) is the integral of sin t from 0 to x. What is g′(0)?

    Answer

  2. Integral from 5 to 5 of any function. Value?

    Answer

  3. An integral from 2 to 7 is 12. What is it from 7 to 2?

    Answer

Stretch Yourself

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

4 problems

  1. Match each part of the theorem with what it does.

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

    • The evaluation part
    • The accumulation part
    • A variable upper limit that is a function of x
    • Computes an integral as F(b) minus F(a)
    • Differentiating an accumulation returns the integrand
    • Requires multiplying by the inner derivative

  2. Integral of 2x from 1 to 3, antiderivative x². Value?

    Answer

  3. The Evaluation

    Integral of 3x² from 1 to 2, with antiderivative x³. What is the value?

    Answer

  4. The Accumulation

    g(x) is the integral of t² from 1 to x. What is g′(3)?

    Answer