Chapter 5 Outline AP Calculus
5.1 Areas and Distances: The main idea is that finding the area between a curve (function) and the x axis
is meaningful. So significant that the whole second part of Calculus AB is about it. This section highlights
that finding the area between a curve and the x axis of a velocity vs. time graph tells you the distance
traveled.
Distances: Lots of applications for this but the favorite for AP is that the area between a curve and the x axis of a
time vs. velocity graph is distance. Look at the image below and consider: distance = velocity*time. So, if you could
find the area of this shape formed by the function you would know the distance
traveled between 0 and 8 seconds.
Rectangles/Trapezoids are used to estimate areas under curves.
When we break up the area under any curve using rectangles or trapezoids and then add up those areas, this is
called taking a Riemann Sum.
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You will need to know how to use Left Hand Rectangles, Right Hand, Midpoint, and Trapezoids.
Know how to find the intervals (n)
Understand Definition 2 on page 365
Know how to work out problems like the Distance Problem on page 367 from a table.
Know how to find areas under curves from a graph.
Be able to answer if you estimate over or under estimates the actual area under the curve.
Understand that as we increase our intervals (the value of n) or rather you are making ∆𝑥 infinitely smaller, this is
a limit. So if you want to find the exact area under the curve then you take a limit as n goes to infinite
𝑛
𝐴𝑟𝑒𝑎 = lim ∑ 𝑓(𝑥𝑖 )∆𝑥
𝑛→∞
𝑖=1
5.2 The Definite Integral: The definite integral IS finding the area under a curve. In this section you learn
about finding the definite integral (the area) by taking the limit of a Riemann sum. This section also
highlights properties of integrals, you should memorize them.
Definite Integrals: In 5.1 you learned that taking the limit of those rectangles/trapezoids got you the area under
the curve. Well this is called a definite integral.
If f is a function defined on an interval [a,b]. We divide the interval into n subintervals of equal width that
we call ∆𝑥. Note that ∆𝑥 =
𝑏−𝑎
.
𝑛
er
We let 𝑥0 = 𝑎 and 𝑥𝑛 = 𝑏 so this gives us the endpoints of the intervals to be 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑖 … , 𝑥𝑛 . This is
important to know in order to understand the notation in Calculus.
So, the definite integral of f from a to b is
𝑏
𝑛
∫ 𝑓(𝑥)𝑑𝑥 = lim ∑ 𝑓(𝑥𝑖 )∆𝑥
𝑛→∞
𝑎
𝑖=1
Properties of Definite Integrals Page 379 Know all the properties of Definite Integrals from page 379 to 381.
Properties 1. – 8. Including the two boxes before #1. On page 379.
5.3 The Fundamental Theorem of Calculus: The fundamental theorem of Calculus establishes a connection
between differential and integral calculus. Differential calculus came from the tangent problem, and integral calculus
arose from the area problem. Basically the integral of a function f(x) (if f is continuous on an interval [a,b]) is a
function g(x) such that g’(x) = f(x).
Don’t forget the + C
See the Fundamental Thm of Calculus Part 1 on page 388. Become a math major and prove it someday 
See the Fundamental Thm of Calculus Part 2 on page 391.
5.4 Indefinite Integrals and Net Change Theorem: Remember that in 5.2 we looked at definite integrals and found
areas now we consider the indefinite integral which is finding the antiderivative. Note: the Net Change Thm is really
just the second part of the Fundamental Theorem.
Don’t forget the +C
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥)
For example: ∫ 𝑥 2 𝑑𝑥 =
1 3
𝑥
3
+𝐶
𝑏𝑒𝑐𝑎𝑢𝑠𝑒
𝑑 1 3
( 𝑥
𝑑𝑥 3
𝑚𝑒𝑎𝑛𝑠 𝐹 ′ (𝑥) = 𝑓(𝑥)
+ 𝐶) = 𝑥 2
Remember that C is a constant.
You should make flashcards and memorize as many of the Indefinite Integrals in the Table on page 398 as you can.
If you know them for the AP test you can just simply use them.
5.5 The Substitution Rule: Commonly referred to as “u substitution” or “u sub.” The substitution rule is a process
that helps us find the integral (antiderivative) of a more complicated function. I will teach you this process but it is
critical that you understand the bigger picture so that you will be able to spot “the second key”.
Example: Evaluate the integral ∫ 3𝑥 2 √1 + 𝑥 3 𝑑𝑥
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The first key to understanding the process I will show you is to understand that if 𝑢 = 𝑓(𝑥) then the
differential (derivative) is 𝑑𝑢 = 𝑓 ′ (𝑥)𝑑𝑥.
The second key to understanding the process is to look at the problem (like the example above) and see if
you can put it into the form ∫ 𝑓(𝑔(𝑥))𝑔′(𝑥)𝑑𝑥. Look carefully, do you see the chain rule: 𝑓(𝑔(𝑥))𝑔′(𝑥).
So now we choose a u. Take notes in class on this example
 The substitution rule for indefinite integrals page 408 and notes in class.
 The substitution rule for definite integrals is on page 411 and notes in class.
 On page 410, u substitution is used to find the integral of tan x. This is another flashcard for your stack!
The integral of tan x is ln |sec x| + C
 Integrals of Symmetric Functions is something that is intuitive if you understand integrals. No need to
memorize the box on page 412 but understand the concept and FEEL free to apply at your convenience.