Math 20A Final Review Outline

Chapter 5: The Integral

Section 5.1: Approximating and Computing Area

• Know how to compute the area under a curve by approximating it with rectangles

• The width of the rectangle is given by

• Know what is meant by summation notation

• Know how to use the right-endpoint, left-endpoint, and midpoint approximations

- Note: f(a +N Δx) is actually just f(b), since

• Know that if f(x) is continuous on [a, b], then the endpoint and midpoint approximations
all approach the same limit L, the true (signed) area under the curve

- If f(x) ≥ 0 on [a, b], then L is the area under the graph of y = f(x) over [a, b]

Section 5.2: The Definite Integral

• Know what a partition P is and how to find the norm of P

• Know what is meant by a Riemann sum

• Know the relationship between the Riemann sum and the Definite Integral

• Know that if a function is continuous on [a, b], then it is integrable over [a, b]
- It is possible to integrate functions that are not continuous, e.g.

• Know what is meant by the signed area of a region. (It is L found in the previous section ,
which is the definite integral

• Know some basic properties of definite integrals:

for any constant C
for all a, b, c


• Know the following integral formulas:

for any constant C

• Know the Comparison Theorem :
If f(x) ≤ g(x) on [a, b] then .
• Know if m ≤ f(x) ≤ M on [a, b], then

Section 5.3: The Fundamental Theorem of Calculus, Part I

• Know The Fundamental Theorem of Calculus, Part I

Assume that f(x) is continuous on [a, b] and let F(x) be an antiderivative of f(x) on
[a, b]. Then then .

• Note: Why do we not have the "+C" like we did for antiderivatives? The answer is that we
do include it, but it cancels out. The antiderivative of f(x) is F(x)+C, but when we evaluate
the integral, we have

- If f(1) = 11, f' is continuous on (−∞,∞), and , what is f(3)?

Section 5.4: The Fundamental Theorem of Calculus, Part II

• Know what is meant by the area function (or cumulative area function):
= signed area from a to x

- Compute the area bounded by the graph y = xl x − 2l, the x-axis, the y-axis, and the
vertical line x = 3.

• Know The Fundamental Theorem of Calculus, Part II
Let f(x) be a continuous function on [a, b]. Then is an antiderivative of
f(x), that is, A'(x) = f(x), or equivalently , . Furthermore, A(x)
satisfies the initial condition A(a) = 0.

- Let f(x) be the continuous function on [−4, 4] whose graph is given below and let

(i) Evaluate F(2), (ii) Evaluate F'(−3), and (iii) Over what intervals shown on the graph
is F concave down? Explain your answer.

- Let . Find the interval(s) on which the graph of F is concave up and
the interval(s) on which the graph of F is concave down.
• Know how to differentiate a function of the form .

- By the FTC, we have that , where F(x) is an
antiderivative of f(x).

- Taking the derivative of this expression (using the chain rule ), we have:

G'(x) = F'(g(x))g'(x) − F'(h(x))h'(x) = f(g(x))g'(x) − f(h(x))h'(x)
- Compute .

Section 5.5: Net or Total Change as the Integral of a Rate

• Know how to compute the net change of a function s(t) over an interval

• Know the difference between displacement and distance traveled:
- Displacement during , where v(t) is velocity

- Total distance traveled during , where v(t) is velocity

• If C(x) is the cost of producing x units of a commodity, then C'(x) is the marginal cost and
the cost of increasing production from a to b is given by

List of Common Mistakes

• Be careful when performing addition with parentheses etc.

- 2(x² + 4) = 2x² + 8 but 2(x² + 4) ≠ 2x² + 8

- (x + y)² ≠ x² + y²

-

- cos(x + y) ≠ cos(x) + cos(y)

- In general, f(x + y) ≠ f(x) + f(y). (The exception is if f(x) is a linear function .)

• Be careful not to make a cancellation error

-
- However, .

• If you take a square root, don't forget the ±

- You may not need the negative term and if that is the case, say why you don't need it.

• Be careful with the property of logs

- ln (a + b) ≠ ln a + ln b, rather, only ln (ab) = ln a + ln b

-

- If you want to simplify something and you don't know if it is valid, plug it into your
calculator (with numbers) and see if they are equal

• When you get an answer, check it to see if it makes sense

- For an exp onential growth problem, if you solve for k and find that it is negative, check
your work. Recall, k > 0 for exponential growth . (It is negative for exponential decay.)

• Don't simplify your answer if you aren't sure that you can

- And if you do, test it to make sure that it does work. If you have any doubt, leave your
answer as it is!

So, You're Stuck on the Test. What Do You Do Now?

First of all, don't panic. If you panic, you will spiral downwards and do poorly. (It has happened
to me. It is not fun.)

You could start screaming at the top of your lungs and wave your arms frantically, but don't.
Rather, relax and take a deep breath. Close your eyes and think about something happy, preferably
not about math. Think of the movie Happy Gilmore (if you have seen it) and Go to your happy
place.

Now, identify what the question is asking for. Write down any formulas that you think apply
from your reference sheets. Also, consider writing down a list of ideas that you will want to try.
If this hasn't led you to the answer, put the question down and move on. See if there are other
problems that you do know how to solve before coming back to it.

Don't spend all your time on one problem. Leave it and come back to it!

Do another problem, maybe you will think about another way to do the previous problem.

If you are confused about what a problem is asking, you can raise your hand and we will try to
help you. We cannot tell you what to do or if you are doing it correctly, so don't ask!

What Can I Do To Prepare?

Get a good night's sleep and eat a good meal before taking the test. Studies have shown that
students do 15-25% better on exams if they get a good night's sleep (at least six uninterrupted
hours of sleep) and have a good meal before their exam.

This means don't pull an all-nighter cramming for this test. Sleep instead. Your mind and
body will thank you for it.

Don't study on the day of the exam, instead just look (brie y) over the material. Your studying
should end the day before. You want to have a night when you can sleep and process everything
that you took in.

If you study the day of the test, you won't remember as much and what you knew will start to
become confused. On the test, you will make silly mistakes and kick yourself later for them. I'm
not kidding, this really does happen. Don't let it be you.