Math 132 Exam 2 Fall 2016• 15 multiple choice questions worth 5 points each.• 2 hand graded questions worth 12 and 13 points each.• Exam covers sections 6.3, 6.5, 7.1-7.5, 7.8, 8.1-8.2• No calculators!• For the multiple choice questions, mark your answer on the answer card.• Show all your work for the written problems. Your ability to make your solution clearwill be part of the grade.Useful FormulasPni=1i =n(n+1)2Pni=1i2=n(n+1)(2n+1)6Pni=1i3=n(n+1)22sin2θ + cos2θ = 11 + tan2θ = sec2θ 1 + cot2θ = csc2θsin(A ± B) = sin A cos B ± sin B cos A cos(A ± B) = cos A cos B ∓ sin A sin Btan(A ± B) =tan A±tan B1∓tan A tan Bsin A sin B =12[cos(A − B) − cos(A + B)]cos A cos B =12[cos(A − B) + cos(A + B)] sin A cos B =12[sin(A + B) + cos(A − B)]sin2x =12(1 − cos 2x) cos2x =12(1 + cos 2x)sin(2θ) = 2 sin θ cos θ cos(2θ) = cos2θ − sin2θRcsc x dx = −ln |csc x + cot x| + CRsec x dx = ln |sec x + tan x| + CMath 132 Exam 2 Page 2 of 111. Suppose you know thatZ30f(x) dx = 12 and the average value of f(x) on [3, 5] is −6.FindZ50f(x) dx.A. 0B. 2C. 3D. 6E. 8F. 9G. 122. Find the average value of f(x) = sin(x)e1−cos(x)on the interval [0, π].A. e2− e−1B.e − e−12πC. e2D.e2− 1πE.e2− 12πF. e2− 1G. −e − 1πMath 132 Exam 2 Page 3 of 113. Evaluate the definite integralZπ120sin2(x) cos2(x) dx.A.π48−164B.π48−√364C.√332D.π96−164E.π96+√364F.π24−112G.π96−√364H.π244. If you are going evaluate the integralZx3√x2− 6x + 13dxusing trig substitution, which substitution should you use?A. x = 3 sin θ + 2B. x =32sec θC. x = 3 tan θ − 6D. x = 2 tan θ + 3E. x = 9 sec θ − 3F. x = 2 sin θ + 3G. x =32tan θH. x = 3 sec θ + 6Math 132 Exam 2 Page 4 of 115. Identify the form of the partial fraction decomposition of the rational function−3x2+ 10x2(x4− 25).A.Ax+Bx + Cx4− 25B.Ax+Bx + Cx2− 5+Dx + Ex2+ 5C.Ax+Bx2+C + Dxx2− 5+E + F xx2+ 5D.Ax+Bx2+Cx − 5+Dx + 5+E + F xx2+ 5E.Ax+Bx2+Cx −√5+Dx +√5+E + F xx2+ 5F.Ax+Bx2+Cxx2− 5+D + Exx2+ 5G.Ax+Bx2+Cx −√5+Dx +√5+Ex2+ 56. Use polynomial long division to write2x4+ x3− 3x2+ 4x − 9x2+ x= Q(x) +p(x)x2+ xwhere Q(x) and p(x) are polynomials. What is p(2)?A. 0B.12C. 1D. 2E. 3F. 4G. 6Math 132 Exam 2 Page 5 of 117. Evaluate the integral:Z1√x2− 16dx.A. arcsinx4+ 4 ln |x +√x2− 16| + CB. lnx4−√x2− 16+ CC. lnx4+√x2− 164+ CD. arcsecx4− 4 lnx +√x2− 16+ CE. lnx +1√x2− 16+ CF.√x2− 164+x4+ CG.23(x2− 16)32+ C8. Evaluate the integralZ√x ln x dxA. CB. ln x + CC.23x3/2+ CD.23x3/2ln x + CE.ln x√x+ CF.23x3/2(ln x −23) + CG.1x+ x3/2+ CMath 132 Exam 2 Page 6 of 119. Evaluate the following improper integral or conclude that it diverges:Z20e√x√xdx.A. e√2B.e√2√2−1√2C. 2e√2√2D.e√2√2+ 1E. 2e√2F. 2(e√2− 1)G. Diverges10. Evaluate the following improper integral or conclude that it diverges:Z∞−∞xx2+ 1dxA. 0B.12C. 1D. 2E. 4F. 6G. 8H. DivergesMath 132 Exam 2 Page 7 of 1111. Compute the following improper integral:∞Z0xe−xdxA. 1B. ∞C. −∞D. 0E. e−1F. e1G. 2e−1H. Diverges12. Let R be the region in the half plane x ≥ 0 bounded by the curvesy = −3x + 3y = x2− 1x = 0Compute the volume of the solid of revolution formed by rotating R about the verticalline x = −1.A. πB.35π6C.3π2D.13π3E.37π5F.35π12G.181π15Math 132 Exam 2 Page 8 of 1113. The region between x = 1 − y2and x = 0 is revolved around the horizontal line y = 3to form a solid. Which of the following integrals represents the volume of the solid?A. 2πZ10(1 − y2)(3 − y) dyB. 2πZ1−1(1 − y2)(3 − y) dyC. 2πZ103(1 − y2)y dyD. 2πZ1−1(1 − y2)y dyE. 2πZ10(3 − x)√1 − x dxF. 2πZ103x√1 − x dxG. 2πZ1−1x√1 − x dx14. Find the length of the curve f(y) = ln(cos(y)) for 0 ≤ y ≤π3.A. ∞B.√3C. 2 +√3D. −ln(√3)E. ln(5√3) − 1F. ln(5√3)G. ln(2 +√3)Math 132 Exam 2 Page 9 of 1115. The curve, y = 4x + 2, between x = 1 and x = 2 is rotated about the x-axis to form asurface. Compute the area of the resulting surface.A. 8√15B. 30π√5C. 32π√17D. 6π√5E. 16√3F. 8√17G. 16π√17Math 132 Exam 2 Page 10 of 11Name:ID:Discussion Section Letter:You can find your discussion section on the front of your exam bookWritten Problem. You will be graded on the readability of your work.Use the back of this sheet, if necessary.16. (a) (7 pts) Find the partial fraction decomposition of16x4− 16.(b) (5 pts) Evaluate the integralZ16x4− 16dx.Math 132 Exam 2 Page 11 of 11Name:ID:Discussion Section Letter:You can find your discussion section on the front of your exam bookWritten Problem. You will be graded on the readability of your work.Use the back of this sheet, if necessary.17. (a) (5 pts) Express the length of the curve y = 2 + exfrom x = 0 to x = 1 as anintegral.(b) (8 pts) Compute the integral from
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