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Polar area and series problems

  1. Apr 5, 2009 #1
    1. The problem statement, all variables and given/known data
    1. Given equations of circles: x^2 + y^2=2 and (x-1)^2 + y^2=1. The graphs intersect at the points (1,1) and (1,-1). Let R be the shaded region in the first quadrant bounded by the two circles and the x-axis.
    (a). Set up an expression involving one or more integrals with respect to x that represents the area of R
    (b). Set up an expression involving one or more integrals with respect to y that represents the area of R.
    (c). The polar equations of the circles are r=sqrt(2) and r=2cos(theta), respectively. Set up an expression involving one or more integrals with respect to the polar angle theta that represents the area of R.

    2.The function f has a Taylor series about x=2 that converges to f(x) for all x in the interval of convergence. The nth derivative of f at x=2 is given f(n) (2)= (n+1)!/3^n for n=>1 and f(2)= 1.

    a. Find the interval of convergence for the Taylor series for f about x=2.
    b. Ket g be a function satisfying g(2) =3 and g'(x)=f(x) for all x. Write the first four terms and the general term of the Taylor series for g about x=2.
    c. Does the Taylor series for g as defined in part (b) converge at x=-2 ?


    2. Relevant equations

    A= 1/2 integral(r^2, dtheta)

    3. The attempt at a solution
    1.
    a. here is what I got: A= integral ( sqrt(2-x^2) - sqrt(1-(x-1)^2) dx) from 0 to sqrt(2). Am I right ?

    b. I got A= integral ( sqrt(2-y^2) - sqrt(1-y^2)-1 dy) from 0 to 1. Am I right ?

    c. I got A = 0.5 * integral ( 2-4*cos^2(theta) dtheta) from -3pi/4 to pi/4. Can you guy check for me ??

    2.
    a. I got the taylor series as 1+(2/3)x+(1/3)x^2+(4/)x^3 +...+ (n+1)!*x^n/3^n.

    Using limit test I get -3<x<3 as the radius of convergence.

    At x=3 I get the series as the sum of (n+1)!, is it a diverge series ?

    b. How should I do this ?
     
  2. jcsd
  3. Apr 5, 2009 #2
    Did I do the first question right ?

    Also, for part b of question 2, do I just use normal Taylor series formula for that or do I have to do another way ?
     
  4. Apr 5, 2009 #3
    For 2c, I found the first four term of the series as: 3+x+x^2/3+x^3/24. Thus, what should be the general term ?

    Do you guys think my general term for part a is right ?
     
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