MATH 2411 ANALYTIC GEOMETRY AND CALCULUS II

FALL 1998 UNIFORM FINAL



1. (4 points each) Consider the region in the xy-plane bounded in the first quadrant by the parabolas

y = x 2 and y 2 = 8x.

  1. SET UP the integral that will give the area of the region.
  2. SET UP the integral that will give the volume of the solid generated by revolving the region about the x-axis.



2. (4 points each) Consider the curve

for 1 < x < 3.

  1. SET UP the integral that will give the arc length of this curve.


  2. SET UP the integral that will give the area of the surface of revolution if the curve is revolved about the x-axis.



3. (6 points) A cylindrical barrel half-full of oil is lying on its side. If each end is circular, with a radius of 3 feet, SET UP the integral that will give the fluid force against one end of the barrel. Assume the density of oil is w = 50 pounds per cubic foot.



4. (3 points each) For the following functions, list the substitution you would make to integrate the function. Find the integral in terms of the new variable. DO NOT INTEGRATE THE NEW INTEGRAL!
INTEGRAL SUBSTITUTION NEW INTEGRAL


    


    


    


    

5. (6 points) The integral can be evaluated using a trigonometric substitution.

  1. List the substitution you would make to integrate the function.
  2. Find the integral in terms of the new variable.
  3. Explain the next step you would make to evaluate the new integral. DO NOT ACTUALLY EVALUATE THE NEW INTEGRAL!
  4. Draw the triangle you would need to get your answer back in terms of x. JUST DRAW THE TRIANGLE, DO NOT EVALUATE THE INTEGRAL



6. (4 points each) Evaluate each of the following integrals.









7. (3 points each) Evaluate the following limits





8. (3 points each)

  1. Determine if the sequence converges. If it converges, find the value it converges to.
  2. Determine if the series converges. State which test you are using and show all your work.

9. (4 points) Find the first 4 terms of the Taylor Series for f(x) = ln( x + 2 ) about the point c = -1.



10. (4 points) Find the radius of convergence of



11. (4 points each endpoint) If the radius of convergence of is R = 1, find the interval of convergence. State which test you are using and show all the work.



12. (3 points each) You are given the series which converges for
-1 < x < 1.

  1. Find the series for
  2. Using part a, find the series for arctan x.



13. (4 points) Let R be the region bounded by the curve r = 1 + cos in polar coordinates. SET UP the integral to find the area of R.



14. (6 points) Let a curve be defined parametrically by x(t) = t 2 - 3t - 4 and y(t) = t 2 - 2t.

  1. What value(s) of t correspond to the point (-4,0)?
  2. Find dy/dx in terms of t.
  3. For what values of t does the graph have a horizontal tangent line?
  4. For what values of t does the graph have a vertical tangent line?

UNIFORM FINAL FOR MATH 2411, FALL 1998: UNIVERSITY OF COLORADO AT DENVER: DECEMBER 1998