Winter 2022 - MATH& 152
Review Sheet
Due March 18, 11:59pm
Directions: the following review sheet contains NO MATHEMATICAL COMPUTATIONS.
Please answer each of the following questions using your notes. You may upload a typed answer,
written over this sheet, or written on a separate sheet of paper to the discussion board. You will be
graded on e↵ort and clarity. Use this to study for your final!
1. For each of the following integrals, write down which of the following techniques should be tried
first to evaluate: Simplify and use Basic Integrals, Simple u-Sub, u-Sub then Simplification,
Integration by Parts, Trig Identities, Trig-Sub, Partial Fraction Decomposition, or Must be
evaluated numerically. Do not solve the integral!
Z
ln x
(a)
dx
x
(b)
Z
p
x x + 1dx
(c)
Z
x2 sin xdx
(d)
Z
x2 + 2x + 1
dx
x4 + 2x2 + x + 2
(e)
Z
p
(f)
Z p
(g)
Z
ex dx
(h)
Z
p
(i)
Z
sin8 xdx
(j)
Z
ex + x2
1
p dx
x+x x
4x2
x4
3
dx
2
x(x + 1)dx
1
dx
1 + x2
1
2. Write down as many basic integration rules we have learned this quarter as you can
2
3. Complete the following simple u-substitution rule:
If F (x) + C is an antiderivative for f (x), then
Z
f (ax + b)dx =
4. State both parts of the Fundamental Theorem of Calculus
5. How did we actually use each part of FTC in this course?
6. There are 3 main interpretations of the integral that we talked about in this course. Fill in the
blanks below to summarize these interpretations:
The integral of f (x) on the interval [a, b]
=
The
under f (x) between
and
=
The
in F on [a, b], if F 0 (x) = f (x)
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7. Write the limit definition of the integral
8. If you were to estimate
(a) Ln
Rb
a
f (x)dx, state the formulas for
(b) Rn
(c) Mn
(d) Tn
(e) Sn
9. If C is an arbitrary constant, explain why each of the following equations are true:
(a) 2 + C = C
(b) ln(2x) + C = ln(x) + C
(c) ex+C = Cex
10. State and describe what “LIPET” means.
11. State the 3 trig-sub forms
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12. Write down all the trig identities we have used in this course
13. Describe the 3 di↵erent rules for decomposing a rational function into partial fractions
14. If you wanted to find the area between 2 functions, use the words “Right,” “Left,” “Top,” and
“Bottom” to describe how to setup the integral
(a) in terms of x
(b) in terms of y
15. If you were finding the volume of a rotational solid where the axis of rotation is vertical, what
method would you be using if you integrated
(a) with respect to x?
(b) with respect to y?
Explain.
16. If f (x) is a function defined on [a, b], and R is the lamina bounded by f (x) with uniform
density ⇢, write the formulas for the following:
(a) the length of f
(b) the surface area of the solid obtained by rotating R about the x-axis
(c) m, Mx , and My
(d) (x̄, ȳ)
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