Past Exam Problems in IntegralsProf. Qiao ZhangCourse 110.202December 6, 2004The following is a list of the problems concerning integrals that appearedin the midterm and final exams of Calc III (110.202) within the last sev-eral years. You may use them to check your understanding of the relevantmaterial. Some other exam problems may b e found athttp://reserves.library.jhu.edu/access/reserves/findit/exams/110/110202.phpNote: These problems do not imply, in any sense, my taste or prefer-ence for our own exam. Some of the problems here may be more (or less)challenging than what will appear in our exam.1. Show that there is no vector field G such thatcurl G = 2xi + 3yzj − xz2k.(Hint: Recall that curl G is the same as ∇ × G.)2. (a) State Green’s Theorem.(b) Use Green’s Theorem to evaluate the contour integralZC(1 + y8) dx + (x2+ ey) dy,where C denotes the boundary of the region enclosed by the curvey =√x and the lines x = 1 and y = 0.3. (a) State the Divergence Theorem. Explain briefly what each symbolin the theorem stands for. (You may assume all the differentiabil-ity you want.)1(b) Use the Divergence Theorem to evaluateRRSF · n dS, whereF(x, y, z) = xyi + (y2+ exz2)j + sin(xy)kand S is the boundary surface of the region E bounded by theparabolic cylinder z = 1 − x2and the planes z = 0, y = 0 andy = 5.4. Let F(x, y, z) = (2x + y2)i + (2xy + 3y2)j.(a) Show that curl F = 0.(b) Find a function f(x, y, z) such that ∇f = F.(c) Use (b) to evaluate the integralRCF · ds, where C is the arc ofthe curve y = sin3x from (0, 0) to (π/2, 1) in the xy-plane.5. Use Gauss’s Divergence Theorem to calculate the total flux out of thecube Ω, given by −1 ≤ x ≤ 1, −1 ≤ y ≤ 1, −1 ≤ z ≤ 1, of the vectorfieldv(r) = 2xyi + (y − y2)j + (x2y + z)k.6. Consider the vector field G(r) = y2zi + (x2y + z2−3z)j + (2yz + ez)k.(a) Use Stokes’ theorem to express the line integralRCG · ds asa surface integral, where C denotes the piecewise linear (square)contour that goes from the origin to (0; 2; 0), then to (0; 2; 2), thento (0; 0; 2), and back to the origin.(b) Hence evaluate the line integral. [HINT: Do not evaluate the lineintegral directly unless you have lots of time and want to checkyour answer.]7. Consider the iterated integralZ2−2Z4y2√xy2ex3dxdy.(a) Sketch the region of integration.(b) Reverse the order of integration, by expressing I as an iteratedintegral with yintegrated first.(c) Hence evaluate I. [WARNING: Do not attempt to evaluate