Review Sheet for Midterm 4


Complex numbers and power series 

There will be a  question on the midterm from the 
material on complex numbers and series. 


Chapter 8

In this chapter various quantities are computed by a method 
which has many applications: approximate the quantity by 
sums which you can interpret as Riemann sums, then compute 
it exactly as the corresponding integral. 

For the test, you are responsible for knowing
integral formulas for various 
such quantities, and how to apply them. 

Formulas: 
-volume by integral of cross section area 
-volume of solid of revolution 
-arc length of graph 
-area of surface of revolution 
-work done by a force on an object 
-center of gravity and moments 
 (for center of gravity: sometimes a coordinate will be 
  obvious by symmetry) 

Not everything is Chapter 8 is a formula problem. For example, 
there work integrals with a different organization. But be 
smart: be ready for the easy (formula) questions. 

Chapter 10 
 
Parametrized Curves. 

Parametrization 
P(t) = (x(t),y(t)) , with the domain of P some interval. 
The set of all points described by P(t) is the curve 
parametrized by this function P. 

View P(t) as the location of a moving particle at time t. 
Know the formula for the velocity (speed) of the particle. 

Be able to graph C. Sometimes this means finding an equation 
satisfied by the t-formulas for x and y (such as y=7x), which tells 
you that the graph of C is part of the corresponding graph 
(such as the graph of y=7x), and then checking to see how much of 
that graph is also the graph of C. 

You should be very familiar with parametrizing motions along 
circles and ellipses, and with angular velocity and period for 
motions on circles. 

There is a very natural formula for distance travelled along 
the path given by P(t) as t goes from a to b.

There is also a formula for the surface area of a surface of 
revolution obtained from a parametrized curve. 

	   SAMPLE EXAMS -- see Testbank 

In some cases (Spring 2002 and Fall 2002 Boyle), 
the  exam 4 is the one related to our material. 
In other cases it is exam 3. 
You can disregard questions on material not related to what 
we covered, e.g.  
Johnson Exam 3 Fall 2000: question 5
Boyle   Exam 3 Fall 1995: questions 1b, 2, 6 
Warner  Exam 3 Fall 2001: Pappus and shell method questions

=============== 

Midterm 4 should be straightforward if you are prepared and 
have done the homework. 
Some integrals might only have to be set up, not computed, to save time. 

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	   Polar Coordinates 

Section 10.4 will not be on Midterm 4. 

For section 10.3 there might be a question. 
Here is a review of polar coordiantes. 

(10.3) Be able to convert between polar and rectangular coordinates for 
points, equations and graphs.  

(10.3) There are a few polar curves with which it is good to be familiar
(n-leaved rose, cardiod, lemniscate, spiral, circles). See p. 656 
for a list. Remark: there are other formulas for giving such  
curves in different orientations (as we saw with the cardioid
r = 1-cos(theta)).  Don't bother with the limacons on p. 656.
Familiarity with such curves might help on the final with a 10.4 
question involving the curves. 

(10.4--for the final) 
Know the formulas for length and area in polar coordinates. 
Be able to find areas of regions bounded between polar graphs 
(examples: 4,5 on pp. 661-662).