Volumes
State the volume V of the solid that lies between x = a and x = b. If the cross-sectional area of S in the plane
Px , through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume V
of S is
We like to nd the
Expression
Substitution
Identity
a 2 x2
a 2 + x2
x2 a2
Table 1: Table of Trigonometric Substitutions
TRIGONOMETRIC SUBSTITUTION
Construct a right triangle dened by sin =
x
a
Construct a right triangle dened by tan =
x
a
Construct a right triangle dened by
TRIGONOMETRIC INTEGRALS
Very Important Identities Needed (Fill in the following)
1. sin(2x) =
2. sin2 (x) =
3. cos2 (x) =
4. tan2 (x) =
Strategy for Evaluating
sinm (x) cosn (x)dx (Try and state in your own words)
Strategy for Evaluating
tanm (x) secn (x)
Substitution
State the Substitution Rule
Whenver you are unsure if your antiderivative is correct, what should you ALWAYS do to check your work?
Work through Example 6 to determine
tan x dx by using the substitution rule. This should nd its way to
your ch
Strategies!
Fill in your table of integration formulae that you should know. Spend 10 minutes a day making sure that you
do.
1
State the 4 strategies in the book and when you would use each one. Try not to copy the book but instead put
them into your own
Series
State the denition of an innite series (just called a series).
What is the partial sum of a series? (Include notation in your answer)
What does it mean for a series to converge?
If a series converges, how do you determine its sum?
Fully dene a geom
Sequences
State the denition and notation for sequences
State the denition of the limit of a sequence.
State the Limit Laws for Sequences
Theorem: If limn |an | = 0, then
Theorem: If limn an = L, and f is continuous at L, then
What does this even mean?
1
PARTIAL FRACTION DECOMPOSITION
Describe the following cases
Case I: The Denominator is a product of distinct linear factors.
Decompose
x2
x+5
+x2
Case II: The Denominator is a product of linear factors, some of which are repeated.
Decompose
x2 + 1
(x 3)(x
Indenite Integrals and the Net Change Theorem
Dene an indenite integral.
Create a table of indenite integrals (page 398)
1
Fully state the Net Change Theorem
What is the dierence between displacement and distance traveled? (This is a COMMONLY missed conce
INTEGRATION BY PARTS
Formula 1: Integration by Parts
Formula 2: Integration by Parts
Work the following problems before class
1.
x cos(x)dx
2.
(x 1) sin(x)dx
3.
ex cos(x)dx
4.
x2 ex dx
5.
tan1 dx
1
The Integral Test
Fully state the Integral Test
What is the p-series?
What is the remainder estimate for the Integral Test? What does it mean?
Determine if the following series converge or diverge:
1.
1
n=2 n ln n
2.
n2
n=3 en
1
Improper Integrals
State the denition of an improper integral of type 1.
State the denition of an improper integral of type 2.
What does it mean for an improper integral to be convergent?
Fully state the Comparison Theorem for improper integrals.
1
Evalua
The Comparison Tests
State the Comparison Test.
What series do we use most often for the purpose of comparison?
1.
2.
When we are comparing our series to a known convergent (or divergent) series, do we really mean for all n or
for some n N for some N ? Wh
Areas Between Curves
State the area A of the region bounded by the curves y = f (x) and the lines x = a, x = b where f and g are
continuous and f (x) g (x) for all x in [a, b]. Sketch an illustration.
The above denition assumes that f (x) g (x) on the who
Alternating Series Test
State the Alternating Series Test
Is the alternating harmonic series convergent?
If you discover that, given an alternating series, limn bn does not go to 0, the series diverges by what test?
Test the following for convergence or d
Absolute Convergence and the Ratio and Root Tests
Dene what it means to be absolutely convergent.
Dene what it means to be conditionally convergent.
We know, as stated in the theorem, that if
an is absolutely convergent, then it is also convergent. Does t
M 408S
Worksheet: Volumes
NO CALCULATORS!
1. Let D be the region enclosed by the curves y = ex , y = 2, and x = 0.
(a) Draw and clearly label the region D.
(b) Find the volume of the solid created when this region is rotated about the y -axis.
(c) . about
M 408S
Worksheet: Completing the square and trig substitution
Practice completing the square:
i. x2 + x
ii. 2x2 + x
iii. 3 3x x2
iv 4x x2
Do the following integrals using trig substitution.
9 x2
1.
dx
x2
1
2.
x3 1 x2 dx
0
2
3.
2
x3
1
dx
x2 1
x
dx
3 2x x2
M 408S
Worksheet: Separable dierential equations
Work your problems on a separate sheet of paper.
1. Find the general solution to each dierential equation.
(a)
(b)
dy
dx
dy
dt
= xey
= 10 2y
(c) (y 2 + xy 2 )y = 1
2. Find the general and particular solutio
M 408S
Worksheet: Practice for exam 2
NO CALCULATORS!
1. Integrate.
(a)
sec4 (x) tan8 (x)dx
(b)
x100 ln(x)dx
(c)
(d)
(e)
dx
x2 + 16
x2 + 2 x 1
dx
x3 x
3
dx
x5
1
2. Let D be the region enclosed by the curves y = sin(x) and y = 0, 0 x .
(a) Draw and clearly
M 408S
Worksheet: More series
1. What does it mean to say that the innite series
an converges?
n=1
2. True or false? For questions like this, true means always true. So, if there is any
situation where the statement is false, the answer is false. For thes
M 408S
Practice quiz: techniques of integration
2/20/2013
Compute the following integrals. Show all work.
x
dx
1 + x2
1.
2.
sin3 (x)dx
3.
x sec2 (x)dx
1
4.
5.
x2
1
dx
+x
x(x2 1)3/2 dx
2
M 408S
Worksheet: The Fundamental Theorem of Calculus
1. Evaluate the following denite integrals. For (b) and (c), draw a picture that illustrates
your answer.
2
x2 x3 dx.
(a)
1
2
cos(x)dx.
(b)
0
2
| cos(x)|dx.
(c)
0
1
(d)
0
1
dx. Hint: trig.
1 + x2
x
f (
E xponential Growth Continuation - Homework
1. A curve passes through the p oint (0, 5) a nd has the property that the slope o f t he curve at every p oint P is twice
the y-coordinate o f P. W hat is the equation o f t he curve?
dy = 2y ~ fdY = f 2dt
dt
M 408S
Worksheet: Comparison of series Answers and explanation
1. Here is the correct order, from slowest to fastest
(o)
(i)
(m)
(p)
(d)
(b)
(c)
(k)
(j)
(l)
(a)
(g)
(f)
(e)
(n)
(q)
(h)
ln(ln n)
ln n
(ln n)100
n0.00001
n
n
100n
n(ln n)1/2
n ln n
n(ln n)2
n
M 408S
Worksheet: Comparison of series
1. All of the following sequences approach as n . Rank them in order from slowest
growing to fastest growing.
(a) n2
(j) n ln n
(b) n
(k) n(ln n)1/2
(c) 100n
(d) n
(l) n(ln n)2
(m) (ln n)100
(e) en
(n) 100n
(f) 2n
(g