Math 230 - Homework 8 Be sure to show all your work and clearly indicate your final answer. Word problems should be answered in complete sentences. Please follow these markers: [C) - You should use a computer to either graph (e.g. on geogebraild) or do calculations I can't expect you to do by hand (e.g. on wolframAlpha). Do not use a computer or calculator on problems without this symbol. Doing so is a violation of academic integrity. [P] - A picture is required to receive full credit for this problem; the picture should be labeled with relevant information. [s) - Your answer needs to be given in the form of a complete sentence, with units where applicable. (*) - This question is optional, but is good practice and might provide some interesting ideas. Be sure to practice your homework, If you struggled with a problem the first time, do it again and you should struggle less. Keep doing the problem until it makes as much sense as the problems you understood the first time around. The homework sets build on each other; many skills practiced in this homework will be called upon in future homework and exams. Do not use any formulas not discussed during class, including any formulas you have learned in physics/chemistry/etc. classes. You will not receive credit for using those formulas. The solutions for all Integrals must include a sketch of the domain. Homework & Learning Objectives: Upon completing this homework, you should feel comfortable with the following skills. . Able to set up and evaluate double integrals in rectangular coordinates . Able to set up and evaluate triple integrals in rectangular coordinates . Able to properly utilize symmetry to simplify integrals Able to use integrals to calculate areas or volumes . Understanding the relationship be densities (e.g. mass density, population density, etc.) and integration . Using an integral to calculate the average value of a function 1. [Double Integrals on Rectangular Regions] Evaluate the following double integrals. Be sure to indicate in your sketch of the region whether you are integrating row-by row or column-by-column. (In some cases, one order of integration will be much easier than the other, so choose wisely.) (a) [ (1 + 2my') dA, where R=0Saga,0 5151 (b) /(6x - ay) + dry) dA, where R=-15 2520 5051 (c) / xsin(zy) dA, where R=05: $20SUS ] 2. [Density and Mass] The density of a thin metal plate on ( $ = $ 3 x 0 S y $ 2 is given by p(x, y) = 2y+ kx], in kg/m . If we want the total mass to be 100 kg, what should we make the value of k? 3. [Double Integrals on General Regions] Evaluate the following double integrals. Be sure to indicate in your sketch of the region whether you are integrating row-by row or column-by-column. (In some cases, one order of integration will be much easier than the other, so choose wisely.) (a) (4y -2x) dA, where E is the rectangular region whose vertices are (1, 0), (1, 3), (2,3), and (2, 0).(b) (2ry + 2) dA, where I is the triangular region whose vertices are (0, 0), (1, 1), and (1, 3). (cy / ry' dA, where It is the region in the trapezoid with vertices (1, 0), (1, 1), (2, 1), and (3, 0). cay Vity dA, where R is the region bounded by r = 0, y = Vi, and y = 1. 4. [Symmetry] Set up the following integrals, and simplify them using symmetry; you do not have to evaluate them. (a) [ (2x' + ay' - day' + bry) dA, where D is the region bounded by y = 5- 1' and y = 1. (by [ (Bye - 2y') dA, where I is the triangular region with vertices (0, -2), (4,0), (0,2). 5. [Double Integrals] For the region R below, determine if the following integrals will be positive, negative, or zero; explain your reasoning. (You don't need to evaluate the integrals to answer this question.) () fu dA (c) | I'dA 6. [Areas] Use a double integral to calculate the following areas. (a) The area of the region bounded by y = 2' and y = r + 2. (b) The area of the region bounded by y = r' and = = y"