Math 241 Midterm exam 3 Spring 2014 Name Section circle one AD1 ADD ADI AD2 ADE ADJ ADA ADF ADK ADB ADG ADL ADC ADH ADM BDA BDF BDK BDP BDB BDG BDL BDQ BDC BDH BDM BDD BDI BDN BDE BDJ BDO READ ALL INSTRUCTIONS CAREFULLY Write legibly and use the boxes for your nal answers where provided Be sure to use correct notation in particular distinguish vectors from scalars by arrow notation use explicit clearly visible dots for dot products etc An answer alone without justi cation will not earn full credit with the exception of the multiple choice Scantron problems 1 2 3 4 7 and 8 If you make a mistake cross out all of your incorrect work We will take points off for incorrect work that is not crossed out even if the correct answer is given elsewhere Problem Point value Test score graded Name section 1 Name NetID 1 Scantron 1 6 Scantron 2 6 Scantron 3 6 Scantron 4 6 Scantron 5 6 6 6 7 6 Scantron 8 6 Scantron Total 13 37 Scantron 50 Math 241 Midterm exam 3 Spring 2014 1 6 points A solid R is composed of material that has density x y z ez and has total mass 2 4 x2 8 x2 y2 ez dz dy dx m 0 0 0 Convert the above integral to an iterated integral in cylindrical coordinates and pencil your answers into the corresponding lines in your Scantron bubble sheet Line 1 Line 2 Line 3 m 0 Line 1 2 A B 2 C D 2 Line 2 A 1 B 2 C 4 D r Line 3 A 4 B 8 C 8 r D 8 r2 Line 4 A er B ez C ez r D ez r2 sin 0 0 Integrand in line 4 dz dr d Math 241 Midterm exam 3 Spring 2014 2 6 points Let E be the portion of the rst octant that is the part of 3 space where x y and z 0 that lies inside the sphere of radius 2 Convert the triple integral 2 x y2 z2 dV E to an iterated integral in spherical coordinates 2 x y2 z2 dV E Line 5 Line 6 Line 7 0 0 Integrand in line 8 d d d 0 Pencil your answers into the corresponding lines in your Scantron bubble sheet Line 5 2 A B 2 C D 2 Line 6 2 A B 2 C D 2 Line 7 A 1 B 2 C 4 D Line 8 A 4 B 2 C 4 2 sin D 4 sin Math 241 Midterm exam 3 Spring 2014 3 6 points Below are two triple integrals written as equivalent iterated integrals in different orders Reverse the order of integration by lling in the missing limits of integration Pencil your answers into the corresponding lines in your Scantron bubble sheet 9 3 x z a Line 11 Line 9 Line 10 f x y z dy dz dx 0 0 0 Line 9 3 A B C D f x y z dx dy dz 0 0 0 Line 10 Line 11 x 3 A B 9 C y D x z 9 3 C 9 D 3 z 2 3 z Line 13 Line 14 Line 12 4 16 z2 16 x2 z2 b z A B f x y z dy dx dz 0 0 Line 12 0 Line 13 A B C D f x y z dz dy dx 0 0 4 16 y2 16 z2 16 x2 z2 Line 14 A B C D 0 4 16 x2 16 z2 16 x2 z2 You may use the space below to sketch the two regions of integration A B C D 4 16 y2 z2 16 x2 y2 16 x2 z2 Math 241 Midterm exam 3 Spring 2014 4 6 points The goal of this problem is to use a transformation change of coordinates to evaluate the double integral x dA R where R is the triangular region with vertices 0 0 3 1 and 2 2 a Under the transformation x 3u 4v y u 4v a region S is transformed to the triangular in your Scantron form ll in which region below corresponds to S region R above In line 15 b From the four choices below select the iterated integral that corresponds to R x dA under the of your Scantron form above change of coordinates and pencil your answer into line 16 A B 1 1 3u 4v du dv 0 1 1 1 u 2 0 12 24u 32v dv du 0 C D 24u 32v dv du 0 0 1 1 u 3u 4v dv du 0 0 0 of your Scantron sheet enter the value of x dA c Using Line 17 R A 7 2 B 10 C 10 3 D 7 6 Math 241 Midterm exam 3 Spring 2014 5 5 1 points 2 a Let F xy sin2 x 2x2 ey Using any valid method compute the line integral F dr C where the closed curve C is the boundary of the plane region D in the right half plane between the circles x2 y2 4 and x2 y2 9 Assume that the curve C is oriented counterclockwise F dr C b How does your answer to part a change if the curve C is oriented clockwise F dr C Math 241 Midterm exam 3 Spring 2014 6 6 points Give parametric equations for the part of the sphere x2 y2 z2 4 that lies below the cone z x2 y2 and above the xy plane You may use any choice of parameters but the domain D must be a rectangle Show your work x D y z Math 241 Midterm exam 3 Spring 2014 7 6 points Consider the surface S that is parameterized by the vector function r u v 1 u2 cos v 1 u2 sin v u where the domain D is given by 1 u 1 and 0 v 2 of a Choose the graph of S given the four choices below and pencil your answer into line 18 your Scantron bubble sheet The x y and z coordinate axes are in their usual positions A B C D b From the choices A E below select the double integral that gives the surface area A S of of your Scantron bubble sheet the above surface S and pencil your answer into line 19 A 1 dA or B ru rv dA or C ru rv dA D or D D ru rv 2 dA D or E D ru rv dA D c Simplify the double integral that computes the surface area A S of S to an iterated integral over a function in the variables u and v From the choices A E below select your answer for of your Scantron bubble sheet A S and pencil it into line 20 2 1 2 1 u2 v2 du dv or B 1 2u2 du dv 0 0 1 1 2 1 2 2 C 1 2u2 du dv or D …
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