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# Review Sheet for MATH 223 at UA3

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 70 views.

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Date Created: 02/06/15
Math 223 Review Some Solutions 1 CHAPTER 16 1 Sketch or describe these nine surfaces assume h is a positive constant Rectangular A z k B y k B 2 h Cylindrical A 6 k B r k C 2 h Spherical A 6 k B p k C t h This was just meant to make you think about the di erent coordinate systems 2 A Evaluate 8 m dxdy 0 yZ Include a sketch of R Answer 2 2 8 278237 i f 3 4 1 1 2 ew dzdy 0 2 Answer You need to change the order of integration rst Then 1 m 1 2 2 2 emdydxxemdxem e17e0e71 0 0 0 C Evaluate W4 1 m2 zc0syd2ddy 0 0 0 Include a sketch of R B Evaluate Include a sketch of R Answer D Evaluate 27r 7r4 3 p2 sin dpd d 0 7r6 0 Include a sketch of W Answer 187Tcos7r6 7 cos7r4 95de W where W is the solid bounded between 2 9 7 x2 7 y2 and zy plane E Evaluate Use cylindrical coordinates 3 27r 9772 r2 cos26 rdzd dr 0 0 0 There is a trick to evaluate fOZW cos2 6d based on 1 cos20sin20 Just do fOZW 1d6 and divide by two since the area between cos26 and sin20 is the same Therefore we get 934716 7T 4 6 39 1 V1712 sinx2 y2dyd 0 m Include a sketch of R 7r4 1 sinr2rdrd6 0 0 F Evaluate Switch to polar then compute 3 Evaluate z V142 cosz2 y2dyd 0 m Again the idea is to switch to polar as above 4 Find the volume of the solid bounded by the surfaces 2y24 zy3 andz0 Use cylindrical coordinates 27r 2 r cos 9r sin 93 rdzdrd6127r 0 0 0 4 2 12 yes dzdy 0 You need to change the order of integration rst 2 12 2 4 1 i 1 yem2x3dydx exem2dz6 0 0 0 2 2 6 Find the volume of the solid between 2 3x2 3y2 and z 12 7 2 7 y2 in the rst octant 5 Evaluate The solids intersect in 2 y2 3 also I am in the rst octant so in cylindrical coordinates 7r2 1242 rdzdrdd 97r2 0 0 372 7 Find the volume of the solid created when a cone 2 xxz y is removed from the upper hemisphere centered at the origin with the radius of V18 You need to do two integrals one for 2 y2 9 and the other one for 2 y2 18 and not in 2 y2 9 8 Find the mass of the solid bounded by a hemisphere with radius of R and a plane if the density at every point is proportional to the distance from the center of the hemisphere This is a little unusual on the escarn you will probably have numbers 9 Find the volume of the solid bounded by the graphs of d2229 y2x7 y07 20 Sketch the solid Actually this is unboundedtry if you add the condition bounded and by the graph ofz 4 10 Find the mass of the region with density 6x7y xy given the region is enclosed by y 7x y d y 17 y 2 This is easy once you sketch the region 2 CHAPTER 17 You can look up the answers to the odd ohes oh lme Number 5 is the some type as the one from class 1 Exercise 15 from section 171 2 Exercise 25 from section 171 3 Exercise 26 from section 172 4 Exercises 15 23 from section 172 5 Explain how you know the following equations parametrize the same line 3 4n 1 20 F Bit 2 117 805 4t 7 5 F12t39 6 Exercises 17 19 from section 173

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