1. Fundamental Theorem of Calculus
Section 5.4
Fundamental Theorem of Calculus

2. Do-Now: Homework Quiz (no calculator)
Estimate 𝑥 2 −2𝑥+3 𝑑𝑥 using….
1) LRAM
2) RRAM
3. Multiple Choice: If you were to estimate (𝑒 𝑥 + 5) 𝑑𝑥 using trapezoids, would your estimate be….?
A) an underestimate B) an overestimate
C) Exactly correct D) not enough info

3. The Fundamental Theorem of Calculus (Part 1)
If f is continuous on [a, b], then the function
𝐹 𝑥 = 𝑎 𝑥 𝑓 𝑡 𝑑𝑡
has a derivative at every point x in [a, b] and
𝑑𝐹 𝑑𝑥 = 𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡=𝑓(𝑥)

4. Example
For 𝐹 𝑥 = 1 𝑥 𝑡 2 −2𝑡+3 𝑑𝑡, compute F’(x).

5. FTC with the chain rule
If the upper limit of integration is a function other than x, use the fact that 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑢 ∙ 𝑑𝑢 𝑑𝑥 .
Find F’(x) for each of the following…
1. 𝐹 𝑥 = 2 𝑥 2 tan 𝑡 𝑑𝑡
2. 𝐹 𝑥 = −1 𝑥 𝑠𝑖𝑛𝑥 𝑡 𝑑𝑡

6. Variable Lower Limit For each problem, find F’(x).
1. 𝐹 𝑥 = 𝑥 12 𝑡 csc 𝑡+5 𝑑𝑡
2. 𝐹 𝑥 = 2𝑥 𝑥 𝑡 dt

7. AP MC

8. AP MC

9. Antiderivative
A function F(x) is an antiderivative of a function f(x) if F’(x) = f(x) for all x in the domain of f. The process of finding an antiderivative is called antidifferentiation.
Name an antiderivative for f(x) = 2x.
Is there more than one?
To name the entire set of antiderivatives, you can refer to it as F(x) + C, where C is an arbitrary constant.

10. Examples
What is the antiderivative for each of the following functions?
1. f(x) = cos x
2. f(x) = sin x
3. f(x) = x3 – 3x2 + 5x – 6
4. f(x) = sec x tan x
5. f(x) = 1/x
6. f(x) = − 𝑥 2
7. f(x) = ex

11. Antiderivatives for Polynomials
For finding the antiderivative of a polynomial, increase the exponent of each term by one and divide by the value of the new exponent.
Examples: Find the antiderivative
1. f(x) = 9x2 – 6x + 7
2. f(x) = 4x4 + 4x3 – 9x

12. Fundamental Theorem of Calculus (Part 2)
If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b], then…
𝑎 𝑏 𝑓 𝑥 𝑑𝑥=𝐹 𝑏 −𝐹(𝑎)
As long as you know an antiderivative, you can use this theorem to find the area between a curve and the x axis.

13. Examples 1. 0 6 [(𝑥 −3) 2 − 3] 𝑑𝑥 2. 1 4 ( 𝑥 − 1 𝑥 2 ) 𝑑𝑥
[(𝑥 −3) 2 − 3] 𝑑𝑥
( 𝑥 − 1 𝑥 2 ) 𝑑𝑥
3. 0 𝜋 sin 𝑥 𝑑𝑥

14. Total Area
Follow these steps to find the total area between a function and the x-axis. This may be necessary if you are given a velocity function and asked to find the total distance travelled (rather than displacement) over a given time interval).
1. Partition [a, b] with the zeros of f.
2. Integrate f over each subinterval.
3. Add the absolute values of the integrals.

15. Example
Assume an object’s velocity (in m/s) can be modeled by the function v(t) = t2 – 6t + 8.
1) Find the change in position (displacement) of the object over the interval [0, 5] seconds.
2. Find the total distance traveled on that same interval.