Math 241 Midterm exam 3 Spring 2014Name:Section (circle one):AD1 AD2 ADA ADB ADCADD ADE ADF ADG ADHADI ADJ ADK ADL ADMBDA BDB BDC BDD BDEBDF BDG BDH BDI BDJBDK BDL BDM BDN BDOBDP BDQREAD ALL INSTRUCTIONS CAREFULLY. Write legibly, and use the boxes for your final answers whereprovided. 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 justification, will not earnfull credit (with the exception of the multiple choice Scantron problems 1, 2, 3, 4, 7, and 8). If you make amistake, cross out all of your incorrect work. We will take points off for incorrect work that is not crossedout, even if the correct answer is given elsewhere.Problem Point value Test score (graded)Name & section 1Name & NetID 1 Scantron1 6 Scantron2 6 Scantron3 6 Scantron4 6 Scantron5 66 67 6 Scantron8 6 ScantronTotal 13 + 37 (Scantron) = 50Math 241 Midterm exam 3 Spring 20141. (6 points) A solid R is composed of material that has density ρ(x, y, z) = ez, and has total massm =∫20∫√4−x20∫8−x2−y20ezdz dy dx.Convert the above integral to an iterated integral in cylindrical coordinates, and pencil your answersinto the corresponding lines in your Scantron bubble sheet.m =∫Line..10∫Line..20∫Line..30Integrand in line..4dz dr dθLine..1 :•..A = −π2•..B =π2•..C = π•..D = 2πLine..2 :•..A = 1•..B = 2•..C = 4•..D = rLine..3 :•..A = 4•..B = 8•..C = 8 − r•..D = 8 − r2Line..4 :•..A = er•..B = ez•..C = ezr•..D = ezr2sin θMath 241 Midterm exam 3 Spring 20142. (6 points) Let E be the portion of the first 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∫∫∫E(x2+ y2+ z2)dVto an iterated integral in spherical coordinates.∫∫∫E(x2+ y2+ z2)dV =∫Line..50∫Line..60∫Line..70Integrand in line..8dρ dθ dϕPencil your answers into the corresponding lines in your Scantron bubble sheet.Line..5 :•..A = −π2•..B =π2•..C = π•..D = 2πLine..6 :•..A = −π2•..B =π2•..C = π•..D = 2πLine..7 :•..A = 1•..B = 2•..C = 4•..D = ρLine..8 :•..A = 4•..B = ρ2•..C = 4ρ2sin ϕ•..D = ρ4sin ϕMath 241 Midterm exam 3 Spring 20143. (6 points) Below are two triple integrals written as equivalent iterated integrals in different orders.Reverse the order of integration by filling in the missing limits of integration.Pencil your answers into the corresponding lines in your Scantron bubble sheet.(a)∫90∫3−√x0∫z0f(x, y, z) dy dz dx =∫Line..90∫Line..100∫Line..110f(x, y, z) dx dy dzLine..9 :•..A = 3 −√x•..B = 3•..C = 9•..D = yLine..10 :•..A = x•..B = z•..C = 3•..D = 9Line..11 :•..A = z•..B = 9•..C = (3 − z)2•..D = 3 −√z(b)∫40∫√16−z20∫√16−x2−z20f(x, y, z) dy dx dz =∫Line..120∫Line..130∫Line..140f(x, y, z) dz dy dxLine..12 :•..A = 4•..B =√16 − y2•..C =√16 − z2•..D =√16 − x2− z2Line..13 :•..A = 4•..B =√16 − x2•..C =√16 − z2•..D =√16 − x2− z2Line..14 :•..A = 4•..B =√16 − y2− z2•..C =√16 − x2− y2•..D =√16 − x2− z2You may use the space below to sketch the two regions of integration.Math 241 Midterm exam 3 Spring 20144. (6 points) The goal of this problem is to use a transformation (change of coordinates) to evaluate thedouble integral∫∫Rx dA,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 triangularregion R above. In line..15 in your Scantron form, fill in which region below corresponds to S.(b) From the four choices below, select the iterated integral that corresponds to∫∫Rx dA under theabove change of coordinates, and pencil your answer into line..16 of your Scantron form.•..A =∫10∫10(3u + 4v) du dv•..B =∫10∫120(24u + 32v) dv du•..C =∫10∫1−u20(24u + 32v) dv du•..D =∫10∫1−u0(3u + 4v) dv du(c) Using Line..17 of your Scantron sheet, enter the value of∫∫Rx dA.•..A =72•..B = 10•..C =103•..D =76Math 241 Midterm exam 3 Spring 20145. (5 + 1 points)(a) Let F = ⟨xy − sin2x, 2x2+ ey2⟩. Using any valid method, compute the line integral∫CF • dr,where the closed curve C is the boundary of the plane region D in the right half-plane betweenthe circles x2+ y2= 4 and x2+ y2= 9. Assume that the curve C is oriented counterclockwise.∫CF • dr =(b) How does your answer to part (a) change if the curve C is oriented clockwise?∫−CF • dr =Math 241 Midterm exam 3 Spring 20146. (6 points) Give parametric equations for the part of the sphere x2+ y2+ z2= 4 that lies below thecone z =√x2+ y2and above the xy-plane.You may use any choice of parameters, but the domain D must be a rectangle.Show your work.x = , y = , z =D ={≤ ≤ , ≤ ≤}Math 241 Midterm exam 3 Spring 20147. (6 points) Consider the surface S that is parameterized by the vector functionr(u, v) =⟨√1 + u2cos v,√1 + u2sin v, u⟩,where the domain D is given by −1 ≤ u ≤ 1 and 0 ≤ v ≤ 2π.(a) Choose the graph of S given the four choices below, and pencil your answer into line..18 ofyour 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) ofthe above surface S, and pencil your answer into line..19 of your Scantron bubble sheet...A =∫∫D1 dA or..B =∫∫D(ru× rv) dA or..C =∫∫D|ru× rv| dAor..D =∫∫D|ru× rv|2dA or..E =∫∫D(ru• rv) dA(c) Simplify the double integral that computes the surface area A(S) of S to an iterated integral overa function in the variables u and v. From the choices..A −..E below, select your answer forA(S), and pencil it into line..20 of your Scantron bubble sheet...A =∫2π0∫1−1√u2+ v2du dv or..B =∫2π0∫1−1(1 + 2u2) du dv or..C =∫2π0∫1−1√1 + 2u2du dv or..D =∫2π0∫√20(1 + u2) du dv or..E =∫2π0∫√20√u2+ v2du dvMath 241 Midterm exam 3 Spring 20148. (6 points) Let F = ⟨2xy, x2, x2+ y2+ ez⟩, and C be the curve given by the intersection ofthe cone z =√x2+ y2and the sphere x2+ y2+ z2= 6,with counterclockwise orientation when viewed from above. Convert the line integral∫CF • dralong the curve C into a surface integral over the surface S that is given by the
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