The Fundamental Theorem of Calculus

The two operations undo each other.

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

What this lesson teaches

The student applies both parts of the Fundamental Theorem of Calculus.

Straightforward practice. Get the method working first.

5 problems

g(x) is the integral of t² from 1 to x. What is g′(4)?
Answer 16
Why 16.
What does the Fundamental Theorem establish?
Answer Differentiation and integration are inverse operations.
Why The two operations undo each other.
Integral of 2x from 0 to 4, antiderivative x². Value?
Answer 16
Why 16 − 0.
Integral of 3x² from 0 to 2, antiderivative x³. Value?
Answer 8
Why 8 − 0.
g(x) is the integral of t³ from 0 to x. What is g′(2)?
Answer 8
Why g′(x) = x³.

The same ideas with more to keep track of.

3 problems

g(x) is the integral of sin t from 0 to x. What is g′(0)?
Answer 0
Why g′(x) = sin x.
Integral from 5 to 5 of any function. Value?
Answer 0
Why F(5) − F(5).
An integral from 2 to 7 is 12. What is it from 7 to 2?
Answer -12
Why Swapping negates it.

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

4 problems

Match each part of the theorem with what it does.
Answer The evaluation part → Computes an integral as F(b) minus F(a); The accumulation part → Differentiating an accumulation returns the integrand; A variable upper limit that is a function of x → Requires multiplying by the inner derivative
Why One part evaluates and the other differentiates.
Integral of 2x from 1 to 3, antiderivative x². Value?
Answer 8
Why 9 − 1.
The Evaluation: Integral of 3x² from 1 to 2, with antiderivative x³. What is the value?
Answer 7
Why 7.
The Accumulation: g(x) is the integral of t² from 1 to x. What is g′(3)?
Answer 9
Why 9.