This exam is closed book. You may use one two-sided sheet of handwritten notes, but no calculators. Please turn o your cell phones.

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1 Math C Midterm 1 Spring 018 Your Name solutions Your Signature Student ID # This exam is closed book You may use one two-sided sheet of handwritten notes, but no calculators Please turn o your cell phones Please place aboxaroundyouranswer to each question In order to receive credit, you must show all of your work If you do not indicate the way in which you solved a problem, you may get little or no credit for it, even if your answer is correct If you need space to work on a given problem, use the backs of the pages and indicate that you have done so Cheating will result in a zero and will be reported to the Dean s Academic Conduct Committee This exam has 5 pages, plus this cover sheet Please make sure that your exam is complete Question Points Score Total 50

2 Math C, Spring 018 Midterm 1 Page 1 of 5 1 (10 points) A planar object occupies the region R of the xy-plane described the inequalities 1 apple x apple 1 and x apple y apple p x The mass density of the object is given by the function (x, y) 1+x The temperature distribution of the object is given by T (x, y) y (a) Sketch R Makesuretoindicateafewsignificantpointsand/ordistancesinyourpicture :::#i :I * E k 's ktz I or - yx > O y1 (b) Express the total mass M of the object as an iterated integral in the order dy dx Do not evaluate the integral Mf I,fyM atxdy (c) Express the average temperature T of the object (weighted by mass) in terms of M and iterated integrals in the order dx dy Do not evaluate the integral a fjof?ryy( 1* ) dxdy + ttnfyyfim#yutxddxdy_)

3 - 1) Math C, Spring 018 Midterm 1 Page of 5 (10 points) Let D be the disc with radius 1 and center at the point (1, 0) of the xy-plane Suppose that D has uniform density 1 (a) Write down an equation describing the boundary circle of D in terms xy-coordinates, then do the same in terms polar coordinates ( + i#f *+yz# y y : (b) Express the moment of inertia of D about the origin as an iterated integral in polar coordinates to (c) Evaluate the integral you found in part (c) The following trigonometric identity may be useful: cos 1(cos( )+1) TERESI ff, r 8 da ftptfr?oa0rlrdrd II r kgo do f IT cos 0 do III ts cos Cuo ) + cos that + g do 0 [ ztzws ( 0 ) + I, sin ( 0 ) +g a ], g it Iµ a ±, ( cos cos Got + 1) tg + st + tz cos (A) t I, tg cos ( a a) + that tz cos g

4 Math C, Spring 018 Midterm 1 Page of 5 (10 points) Set up the following integrals in the specified orders of integration Do not evaluate the integrals In this problem you only need to show your final answer (a) Z 1 Z x Z y Z ' Z Z f(x, y, z)dzdydx o it ty f(x, y, z)dx dz dy ii e Ia ftpt#kieisiieeeeneee (b) Let E be the region of the first octant that lies inside the sphere x + y + z and below the plane z y ZZZ Z;R Z Z f(x, y, z)dzdydx out f(x, y, z)dxdzdy E yxty +- none + fee FOKTT fomtefi/uyiz)dxdzdy - As announced during the exam, :* n order were accepted as correct ME#FzEyttyIFIF//fotfl*fou*fcxiyzidxdy

5 Math C, Spring 018 Midterm 1 Page of 5 (10 points) Consider the region E that lies above the cone z p x + y and inside the sphere x + y +(z 1) 1 (a) Sketch R Makesuretoindicateafewsignificantpointsand/ordistancesinyourpicture tfe E t * tr ; Cone : (b) Find equations in spherical coordinates defining the cone and the sphere that bound E Your equations should be as simple as possible poos $Rpsin$wso)+(psin$sino p sin 01 cos sing ) Sphere : +yt z Zz pp_os$ ; pcos0_ (c) Express the volume of E as an iterated integral in spherical coordinates Do not evaluate vdiekfjyfto/pylpsinoldpdo -

6 Math C, Spring 018 Midterm 1 Page 5 of 5 5 (10 points) Consider the double integral ZZ R x y +1 dx dy, x + y where R is region of the xy-plane bounded by the lines x y 1, x y, x + y 1 and x + y (a) Find new variables u u(x, y) andv v(x, y) intermsofwhichtheintegrandandthe region in double integral above become simpler Sketch the region in the uv-plane that corresponds to R C- ) Ctr nv ) :(--1) flit ) F±±I EEIIt (b) Express x and y in terms of u and v, and compute the Jacobian y) v) *±Et oin#aetfeae*dv-uys/yttutdetftgtsfatchftf (c) Compute the double integral above by changing the variables of integration to u and v ftpfe#ytdxdyfifizw*tsdudotssitoim*iidrtlott5tttitattztdtist

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.

Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral