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Practice Exam 3 (1770373)
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7.3 Volume: The Shell Method
p+
w
2
p
p
w
2
p is the distance from the axis to the center of the region.
w
2
w
Radius of outer cylinder: p +
2
Radius of inner cylinder: p
V = (volume of cylinder) (volume of hole)
= p + h p h = 2phw = 2(average radius)(h
7.4 Arc Length and Surfaces of Revolution
Rectifiable curve: a curve that has a finite arc length.
Definition of Arc Length:
Let the function given by y = f(x) represent a smooth curve on the interval
[a, b]. The arc length of f between a and b is:
s=
b
a
7.2 Volume: The Disk Method
Consider a cylinder.
We know a formula to find the volume of this shape. How could
we find the volume if we did not know a formula? One approach
would be a method called the disk method. To find the volume,
consider how we coul
7.1 Area of a Region Between Two Curves
b
[ f (x ) g (x )]dx
a
b
=
f ( x)dx
b

a
g ( x)dx
a
Area of a Region Between Two Curves
If f and g are continuous on [a, b] and g(x) < f(x) for all x in [a, b], then
the area of the region bounded by the graphs
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Practice Exam 3 (1770373)
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7.5 Work
Work is done by a force when it moves an object.
Definition of Work Done be a Constant Force
If an object is moved a distance D in the direction of an applied constant force F,
then the work W done by the force is defined as W = FD
Ex 1 Determine
7.6 Moments, Centers of Mass, and Centroids
Mass: a measure of a bodys resistance to changes in motion, and is
independent of the particular gravitational system in which the body is
located.
Some people inaccurately equate mass to weight.
Force = (mass)(
8.4 Trigonometric Substitution
If you have a radical of the type: a 2 u 2 , a 2 + u 2 , or u 2 a 2 , where a is a
constant and u is a function, you can make substitutions.
Recall: cos 2 = 1 sin 2 , sec 2 = 1 + tan 2 , and tan 2 = sec 2 1 .
If a > 0 , let
8.5 Partial Fractions
Another technique to integrate a rational function is to decompose the rational
function into smaller rational functions which are easier to integrate.
Decomposition of N(x)/D(x) into Partial Fractions
1. Divide if improper: If N(x)/
8.3 Trigonometric Integrals
Recall the following Trigonometric Identities:
sin2x + cos2x = 1
1 cos 2 x
sin 2 x =
2
1 + cos 2 x
cos 2 x =
2
Guidelines for Evaluating Integrals Involving Sine and Cosine
1. If the power of the sine is odd and positive, save
8.2 Integration by Parts
Integration by Parts: a technique that can be applied to a variety of functions,
particularly useful in integrands involving products of algebraic functions.
Theorem 8.1 Integration by Parts
If u and v are functions of x and have
7.7 Fluid Pressure and Fluid Force
Pressure: The force per unit of area over the surface of body.
Definition of Fluid Pressure:
Pressure on an object at depth h in a liquid is
Pressure = P = wh
where w is the weightdensity of the liquid per unit of volum
8.1 Basic Integration Rules
Ex 1 Select the basic integration formula you can use to find the integral,
and identify u and a when appropriate.
a) (3 x 2 )4 dx
b)
t
2
2t 1
dt
t + 2
Ex 2 Select the basic integration formula you can use to find the integral,