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Cogito

Calculus · Chapter 8 · Lesson 1

Practice Worksheet

The Fundamental Theorem of Calculus

The two halves of the subject turn out to be one.

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

NameDate

Figure — use these to answer the problems

The shaded area between the two ends is what F(b) − F(a) computes.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x + 0
  • Point(0, 0)
  • Second point(3, 9)
  • Slope between them3
a1
b0
c0
From x0
To x3
Figure 1

Warm-Up Review

From earlier lessons. Loosen up before the new work.

2 problems

  1. From “Riemann Sums”

    From 0 to 12 with 6 rectangles. Width of each?

    Answer

  2. From “Antiderivatives”

    Antiderivative of 8x is kx². What is k?

    Answer

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. Integrate 2x from 2 to 5. Answer?

    Answer

  2. What does the Fundamental Theorem connect?

    • Differentiation and integration, as inverse processes.
    • Limits and continuity.

  3. Integrate 2x from 0 to 5. Answer?

    Answer

  4. Integrate 2x from 1 to 3. Answer?

    Answer

Build It Up

The same ideas with more to keep track of.

2 problems

  1. Integrate 3x² from 0 to 2. Answer?

    Answer

  2. An antiderivative of 4x³ is x to what power?

    Answer

Stretch Yourself

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

2 problems

  1. The Integral

    Integrate 2x from 0 to 4. What is the answer?

    Answer

  2. The Antiderivative

    The derivative of x³ is 3x². So an antiderivative of 3x² is x to what power?

    Answer