Polar area and series problems

[nns91]

Homework Statement

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 ?

Homework Equations

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

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 ?

 

[nns91]

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 ?

 

[nns91]

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 ?