Unit: The Basics of Integration
Module: The Fundamental Theorem of Calculus
Approximating Areas of Plane Regions
The two key questions of calculus have a subtle connection.
When trying to find the area of a complicated region, try approximating the area
w
AP Calculus
Chapter 3 Review
1. An object starts at the origin and moves along the x-axis with velocity:
v(t ) = 20t t 2 , 0 t t10
a. What is the position of the object at any time t, 0 t t10 ?
b. When would the object reach its maximum velocity and what
Finding Volumes Using Cross-Sectional Slices
The volume of a solid with vertical cross-sections of area A(x) is V where:
V =
b
A( x ) dx .
a
The volume of a solid with horizontal cross-sections of area A(y) is V where:
V =
b
A( y ) dy .
a
Finding the volu
Center of Mass
The center of mass of an object is the point where you can assume all the mass is
concentrated.
A moment can be thought of as a systems tendency to rotate about an axis.
N
The moment about the y-axis is
mn xn .
n =1
N
The moment about the
AP Calculus
Chapter 3 Review Solutions
1. An object starts at the origin and moves along the x-axis with velocity:
v(t ) = 20t t 2 , 0 t t10
a. What is the position of the object at any time t, 0 t t10 ?
v(t )dt = distance
t
20 2 t 3
t3
2
t
= 10t
2
30
Finding the Average Value of a Function
The average value of a function on an interval is the area under the curve divided
by the length of the interval.
Average value =
1
ba
b
f ( x ) dx .
a
Normally to find an average, you would just add up
all the valu
Chapter 2 Review
AP Calculus C
Denitions
Denition 1. Antidierentiation is the process or operation that reverses dierentiation. You may
also use the term indenite integration as an alternate name for this process.
Denition 2. A function F is an antideriva
Converting Radicals into Trigonometric Expressions
Consider the square root of the sum or difference of two squares as information
about an unknown right triangle. Trigonometric substitution allows you to convert
the square roots into less complicated tri
Introduction to Integration by Parts
Integration by parts reverses the product rule.
Integration by parts requires a product of two functions: one with a simple
derivative and one with a simple integral.
The integral of lnx is a tough nut to crack. The
te
Chapter 6 Review
AP Calculus C
Denitions
Denition 1. A sequence is a pattern of numbers usually denoted as
cfw_an = cfw_a1 , a2 , a3 , . . ..
Denition 2. For a real number L, the limit of a sequence cfw_an is L, that is
lim an = L.
n
Sequences that have
The First Type of Improper Integral
An improper integral is a definite integral with one of the following properties: the
integration takes place over an infinite interval or the integrand is undefined at a
point within the interval of integration.
With s
An Overview of Trig Sub Strategy
Use trigonometric substitution to evaluate integrals involving the square root of
the sum or difference of two squares.
1. Match the square root expression with the sides of a right triangle.
2. Substitute the correspondin
Solids of Revolution
Revolving a plane region about a line forms a solid of revolution.
The volume of a solid of revolution using the disk method where R(x) is the radius
of the solid of revolution with respect to x is V where:
V =
b
[ R ( x )] 2 dx .
a
S
Introduction to Arc Length
Arc length is the length of the curve.
The arc length of a smooth curve given by the function f (x) between a and b:
b
1 + [f ( x )]2 dx .
a
When measuring how long a line is, you can just
use a ruler or the distance formula. Bu
Introduction to Integrals with Powers of Sine and Cosine
Use a double-angle identity to integrate sin2 x or cos2 x.
Use u-substitution to integrate sin3 x cos x.
A double-angle identity is a trigonometric identity
where the argument of the trigonometric f
Unit: The Basics of Integration
Module: Numerical Integration
Deriving the Trapezoidal Rule
The trapezoidal rule approximates the area A of the region bound by the curve of
a continuous function f (x) and the x-axis using N partitions on [a,b].
The trapez
AP Calculus
Chapter 2 Review Solutions
1. Integrate the following.
dx
a.
x 1
b.
x cot x dx
2
6x2
dx
c.
(5 x3 )
d.
1
1
dx
1 x2
1
1
dx
e.
0 1 + x2
0
ln ( x 1 ) + C
du 1
1
= cot udu = ln sin x 2 + C
2 2
2
2
u = x du = 2 xdx
cot u
6
6 du
(5 x ) x dx = u
Integrating Composite Trigonometric Functions by Substitution
Integration by substitution is a technique for finding the antiderivative of a
composite function. A composite function is a function that results from first
applying one function, then another
Integrals of Other Trigonometric Functions
Integrate tangent and cotangent by expressing them in terms of sine and cosine
and then using u-substitution.
Integrate secant and cosecant by multiplying by an expression equal to one.
Although many trigonometri
AP Calculus
Chapter 2 Review
Chapter homeworks are open-book assessments. You may refer to the course notes, supplementary notes,
and/or the lectures. You may also contact your instructor for additional help. You are allowed to use a calculator.
You DO NO
AreaProblem
As noted in the first section of this section there are two kinds of integrals and to this point weve
looked at indefinite integrals. It is now time to start thinking about the second kind of integral :
Definite Integrals. However, before we d
Introduction to Work
Work is the energy used when applying a force over a distance.
For a constant force F, work is the product of the force and the change in distance.
For a changing or variable force F(x) on [a ,b ], work is given by the integral:
b
F (
Gravity and Vertical Motion
The acceleration due to gravity of an object is a constant.
Given the initial velocity and initial position of an object moving vertically, you
can use the fact that the acceleration due to gravity is a constant to find the
vel
Area Between Two Curves
The definite integral can be used to calculate the area between a curve and the xaxis on a given interval.
To find the area of a region bounded by the graphs of two functions, take the
definite integral of the difference of the two
Chapter 3 Review
AP Calculus C
Denitions
Denition 1. The position function tells us where an object is at a given point in time.
Denition 2. The velocity is the rate of change of the position with respect to time.
Denition 3. The acceleration is the rate
Name:
110.107 CALCULUS II (Biology & Social Sciences)
FALL 2010
MIDTERM EXAMINATION
October 12, 2010
Instructions: The exam is 7 pages long, including this title page. The number of points each problem
is worth is listed after the problem number. The exam
Name:
Section Number:
110.107 CALCULUS II (Biology & Social Sciences)
FALL 2010
MIDTERM EXAMINATION
December 1, 2010
Instructions: The exam is 7 pages long, including this title page. The number of points each problem
is worth is listed after the problem
Calculus II
Fall 2009
Exercises For Exam II
1.
Give a summary of partial derivatives, tangent planes, linearization,
Summarize how to nd max/min.
Give a summary of system of linear equations. (How to solve, Stability)
Give a summary of counting.
2. Fi
FINAL PRACTICE EXAM IV
YI LI
1. Let f (x, y ) = x2 y 2 with constraint function
2x + y = 1.
Using Lagrange multipliers to nd all extrema.
Write g (x, y ) = 2x + y 1. Then
f (x, y ) =
The equation
2x
,
2y
g (x, y ) =
2
1
f (x, y ) = g (x, y ) implies
2y =
Math 107 Practice Exam 1
Part I
1. (10pts) Evaluate
2. (10pts) Evaluate
x3 ln x dx
1
x2 4x+3
dx
3. (10pts) Two mg of radioactive material decays to 1.3 mg after 10 days.
Find the half-life of the material.
4. (15pts) Find f (x) if f (x) = 2 f (x) + 4 ,
f