Quiz 3-sol - Math 116 Name Quiz 3 Evangelos Dimou Section Note Show all your work No calculators allowed 1(4 points Let D be the region bounded by x = y

Quiz 3-sol - Math 116 Name Quiz 3 Evangelos Dimou Section...

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Math 116 Quiz 3 January 26, 2017 Name: Evangelos Dimou Section: 45/65/85 Note: Show all your work. No calculators allowed. 1. (4 points) LetDbe the region bounded byx=y2andx= 4.(a) Set up an integral that describes the volume of the solid whose base is theregionDand whose cross-sections perpendicular to they-axis are squares.You do notneed to compute the integral.(b) Find an expression that equals the perimeter of the regionD. You do notneed to compute any integrals in your expression. Solution y x x = y 2
2. (6 points) The graph of the functionf(x) =-x3+xis given below. You do notneed to compute any integrals in this problem.(a) Set up an integral that equals the area of the region bounded between thegraph offand thex-axis.(b) Set up an integral that describes the volume of the solid that is obtainedafter the shaded region is rotated abouty=-2.(c) Set up an integral that describes the volume of the solid that is obtainedafter the shaded region is rotated about they-axis. y x y = f ( x ) Solution (a) We check thatf(x) = 0⇔ -x3+x= 0⇔ -x(x2-1) = 0, that is,x= 0,±1. ThereforeA=Z1-1|f(x)|dx=Z-1x3-x dx+Z1-x3+x dx(= 2Z1-x3+x dx)(b) The volume of a washer would be ΔV=π[(-x3+x+ 2)2-(-2)2] Δx, soV=Z1π[(-x3+x+ 2)2-22]dx(c) The volume of a cylindrical shell would be ΔV= 2πx(-x3+x) Δx, soV=Z12πx(-x3+x)dx

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