Study Guide for Test #3MA 242 on campus, MA 242.601 and MA 242.651The 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 sphericalcoordinates.1. Chapter 12, section 1: Double integrals over rectangles(a) You should know the general definition (5) of the double integral of a functionf(x,y) over a rectangular region R.(b) You should know the definition at the top of page 842 for the volume b elow thegraph of a function and ab ove a region R in the xy-plane.(c) You should know the definition of the average value of a function on a rectanglegiven on page 844.(d) You should know the properties of double integrals given on page 847 of your textbook.(e) You should realize that the above items will be generalized in section 12.3 wherewe 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 inproblems 3 - 10 on page 853, and problems 3 - 6 on page 890. At most you willbe required to use ”substitution” to evaluate such integrals.(b) You should know Fubini’s theorem ((4) - page 850) and be able to apply it toproblems 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 evaluatingdouble integrals over type I and type II regions.(b) You should be able to decompose a general region into a set of subregions, eachof 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) andabove 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+y2and z = 18− x2− y2would be found as follows. These two paraboloids intersect in the circle of radius9 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 insidethis circle be denoted D. Then the volume between the two paraboloids would begiven by the double integral of 18 − x2− y2over D MINUS the double integral ofx2− y2over the region D.1(e) You should be able to ”reverse the order of integration” on a given iterated inte-gral. To do this you:i. Use the limits on the given iterated integral to write down the description ofthe 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 torewrite 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 coordinatesi. You should know and be able to use theorems (2) and (3) to set up and eval-uate double integrals in polar coordinates. This involves using the trasforma-tion equation x = r cos(θ) and y = r sin(θ). See the e xamples worked in thetextbook and the examples I worked for you in class.(g) Chapter 12, section 5: Applications of double integrals. Below are the applicationsyou are responsible for.i. Volume below the graph of a function and above a general region in thexy-plane.ii. Average value of a function over a general region in the xy-planeiii. Area of a general region in the xy-plane: In this case the integrand of thedouble integral will be 1.iv. Densities: If ρ(x, y) is the mass (or charge) density of a region D in the xy-plane, then the total Mass (or charge) of the region is the double integral ofρ(x, y) over the region D.(h) Chaper 12, section 7: Triple integrals in Cartesian coordinates.i.ii. You should know and b e able to apply the three versions (6), (7) and (8) ofFubini’s theorem for triple integrals. See the examples worked in the bookand the many exmaples I worked for you in class.iii. Know the formula (12) for the volume of the 3-dimensional region using tripleintegration.4. Chapter 12, section 8: Triple integrals in c ylindrical 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(θ), y = r sin(θ)and z = z and Spherical: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ) and z = ρ cos(φ)in order to be able to transform an integrand given in terms of x,y and z to eitherof these coordinate systems.(c) Be able to use the above to set up a triple integral as a triple iterated integral ineither cylindrical or spherical coordinates. See the examples worked in the textbook and the many examples I worked for you in