Cogito
Calculus · Chapter 8 · Lesson 1
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
What this lesson teaches
The student states and applies both parts of the Fundamental Theorem of Calculus.
- Differentiation and integration undo each other.
- The definite integral from a to b is F(b) − F(a).
- An infinite sum is computed by subtracting two numbers.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Riemann Sums: From 0 to 12 with 6 rectangles. Width of each?
Answer 2
Why 2.
Review — Antiderivatives: Antiderivative of 8x is kx². What is k?
Answer 4
Why 4.
Warm Up
Straightforward practice. Get the method working first.
4 problemsIntegrate 2x from 2 to 5. Answer?
Answer 21
Why 21.
What does the Fundamental Theorem connect?
Answer Differentiation and integration, as inverse processes.
Why They undo each other.
Integrate 2x from 0 to 5. Answer?
Answer 25
Why 25 − 0.
Integrate 2x from 1 to 3. Answer?
Answer 8
Why 9 − 1.
Build It Up
The same ideas with more to keep track of.
2 problemsIntegrate 3x² from 0 to 2. Answer?
Answer 8
Why x³: 8 − 0.
An antiderivative of 4x³ is x to what power?
Answer 4
Why Derivative of x⁴ is 4x³.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Integral: Integrate 2x from 0 to 4. What is the answer?
Answer 16
Why 16.
The Antiderivative: The derivative of x³ is 3x². So an antiderivative of 3x² is x to what power?
Answer 3
Why x³.