Fundamental Theorem of Calculus

Estimate
◦ 1) LRAM
◦ 2) RRAM


𝑥 2 − 2𝑥 + 3 𝑑𝑥 using….
3. Multiple Choice: If you were to estimate
4 𝑥
(𝑒 +
0

4
0
5) 𝑑𝑥 using trapezoids, would your
estimate be….?
A) an underestimate B) an overestimate
C) Exactly correct D) not enough info



If f is continuous on [a, b], then the
function
𝐹 𝑥 =
𝑥
𝑓
𝑎
𝑡 𝑑𝑡
𝑥
𝑓
𝑎
𝑡 𝑑𝑡 = 𝑓(𝑥)
has a derivative at every point x in [a, b]
and
𝑑𝐹

𝑑𝑥
=
𝑑
𝑑𝑥

For 𝐹 𝑥 =
𝑥
1
𝑡 2 − 2𝑡 + 3 𝑑𝑡, compute F’(x).

If the upper limit of integration is a function
other than x, use the fact that



𝑑𝑦
𝑑𝑥
=
𝑑𝑦
𝑑𝑢
Find F’(x) for each of the following…
1. 𝐹 𝑥 =
𝑥2
tan 𝑡
2
2. 𝐹 𝑥 =
𝑥 𝑠𝑖𝑛𝑥
−1
𝑑𝑡
𝑡 2 + 1 𝑑𝑡
∙
𝑑𝑢
.
𝑑𝑥

For each problem, find F’(x).

1. 𝐹 𝑥 =
12
𝑡 csc
𝑥
2. 𝐹 𝑥 =
𝑥2
2𝑥

𝑡 + 5 𝑑𝑡
𝑡 2 + 1 dt




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.






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
1

6. f(x) =

7. f(x) = ex
1 − 𝑥2

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





If f is continuous at every point of [a, b], and
if F is any antiderivative of f on [a, b], then…
𝑏
𝑓
𝑎
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.



1.
6
[(𝑥
0
2.
4
(
1
3.
𝜋
sin 𝑥
0
− 3)2 − 3] 𝑑𝑥
𝑥 −
1
)
𝑥2
𝑑𝑥
𝑑𝑥




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.



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.