Calculus with Analytic Geometry 2 and Calculus for Engineers 2 Final Examination August 5, 2011

No graphing calculators or cell phones. Show your work to ensure full credit. Name: time out______time in______

  1. [10]Find the volume of the solid generated when the area enclosed by the curves y = x3 and y = x2 is rotated about the x–axis.
  1. [10]A heavy rope, 60 feet long, weighs one pound per foot, and hangs over the edge of a building 120 feet high. Find the work done when the rope is pulled to the top of the building.
  1. [7]Find .
  1. [7]Find
  1. [7]Find
  1. [7]Find
  1. [7]Integrate
  1. [7]Integrate
  1. [7]Integrate
  1. [10]Use Simpson’s Rule with n = 6, to approximate the integral, to six decimal places.
  1. [8]Determine whether the integral is convergent or divergent. Evaluate it if it is convergent:
  1. [10]Find the length of , from y = 1 to y = 2.
  1. [10]Find the area of the surface obtained by rotating the curve, about the y-axis.
  1. [12]Find the centroid of the region bounded by the curves,
  1. [10]Find the area enclosed by
  1. [10]Find the area enclosed by the astroid,

In problems 17-24, tell whether the series is AC, CC or D. Give reasons.

  1. [7]
  1. [7]
  1. [7]
  1. [7]
  1. [7
  1. [8]Find the sum of .
  1. [8]Find the radius of convergence and interval of convergence of .
  1. [8]Determine a power series representation for the function and determine the interval of convergence.
  1. [8]Find the Maclaurin series for f(x)= x sin 2x and give its radius of convergence.
  1. [8]Find the Maclaurin series forf(x)= xsin 2x and give its radius of convergence.
  1. [8]Find the Taylor series for x3 - 1 about -2.

If the area A of a region is given by an integral , then the moment µx about the x-axis is , the moment µy=, and the centroid is (µy/A, µx/A).