Study Guide for Test 3 MA 242 on campus MA 242 601 and MA 242 651 The test will cover the following sections of Chapter 12 1 2 3 4 5 see below 7 and 8 In addition it will contain material from Chapter 9 section 7 on cyclindrical and spherical coordinates 1 Chapter 12 section 1 Double integrals over rectangles a You should know the general definition 5 of the double integral of a function f x y over a rectangular region R b You should know the definition at the top of page 842 for the volume below the graph of a function and above a region R in the xy plane c You should know the definition of the average value of a function on a rectangle given on page 844 d You should know the properties of double integrals given on page 847 of your text book e You should realize that the above items will be generalized in section 12 3 where we will no longer require the region of integration to be a rectangle 2 Chapter 12 section 2 Iterated integrals and Fubini s Theorem a Be able to compute iterated integrals double and triple such as those given in problems 3 10 on page 853 and problems 3 6 on page 890 At most you will be required to use substitution to evaluate such integrals b You should know Fubini s theorem 4 page 850 and be able to apply it to problems like those worked in the text and the examples I worked for you in class 3 Chapter 12 section3 Double integrals over general regions a You should know how to use the two basic theorems 3 and 5 for evaluating double integrals over type I and type II regions b You should be able to decompose a general region into a set of subregions each of type I or type II See problems 41 and 42 at the end of the section c You should be able to compute volumes below graphs of functions f x y and above a general region in the xy plane d You should be able to find the volume between the graphs of two functions f x y and g x y by reducing this problem to one you have already solved For example the volume of the region between the paraboloids z x2 y 2 and z 18 x2 y 2 would be found as follows These two paraboloids intersect in the circle of radius 9 centered on the origin in the xy plane You find this curve of intersection by solving the two equations simultaneously eliminating z Let the region inside this circle be denoted D Then the volume between the two paraboloids would be given by the double integral of 18 x2 y 2 over D MINUS the double integral of x2 y 2 over the region D 1 e You should be able to reverse the order of integration on a given iterated integral To do this you i Use the limits on the given iterated integral to write down the description of the region in set notation ii Use the set notation to sketch the region iii If the set notation indicates that the region is type I then use the sketch to rewrite it as type II and conversely iv Set up the iterated integral in the opposite order f Chapter 12 section 4 Double integrals in polar coordinates i You should know and be able to use theorems 2 and 3 to set up and evaluate double integrals in polar coordinates This involves using the trasformation equation x r cos and y r sin See the examples worked in the textbook and the examples I worked for you in class g Chapter 12 section 5 Applications of double integrals Below are the applications you are responsible for i Volume below the graph of a function and above a general region in the xy plane ii Average value of a function over a general region in the xy plane iii Area of a general region in the xy plane In this case the integrand of the double integral will be 1 iv Densities If x y is the mass or charge density of a region D in the xyplane then the total Mass or charge of the region is the double integral of x y over the region D h Chaper 12 section 7 Triple integrals in Cartesian coordinates i ii You should know and be able to apply the three versions 6 7 and 8 of Fubini s theorem for triple integrals See the examples worked in the book and the many exmaples I worked for you in class iii Know the formula 12 for the volume of the 3 dimensional region using triple integration 4 Chapter 12 section 8 Triple integrals in cylindrical coordinates a See Chapter 9 section 7 for the definition of cylindrical and spherical coordinates b Know the transformation equations Cylindrical coords x r cos y r sin and z z and Spherical x sin cos y sin sin and z cos in order to be able to transform an integrand given in terms of x y and z to either of these coordinate systems c Be able to use the above to set up a triple integral as a triple iterated integral in either cylindrical or spherical coordinates See the examples worked in the text book and the many examples I worked for you in class 2