Calculus 2

1. Evaluate each improper integral or show that it diverges.

(1)  +

1 x

dx



(2)  +

− ++ 22 2

xx

dx

(3)  −

2

0 2)1( x

dx

(4)  −

e

xx

dx

1 2 )(ln1

2. For what values of k does the integral )( )(

ab ax

dxb

a k 

−  converge and for what

values does it diverge?

3. Calculate the area of the following regions.

(1) The region bounded by xx eyey −== , and 1=x .

(2) The region bounded by )cos1(2  += a (a>0).

4. Sketch and find the area of the region bounded by  cos3= and

 cos1+= .

5. Find the volume of the solids generated by revolving about the x-axis the

region bounded by 02 =− yx and the parabola xy 42 = .

6. Find the length of 3

4 xy = between x = 1/3 and x = 5.

7. Find the length of the curve 41;12,23 32 −=+= ttytx .

8. Fill the blanks.

(1) If  

=

 

  

 −

1 2

3

n

n

n

x a converges at 0=x , then it __________ (converges, diverges)

at 5=x .

(2) If 2lim 1

= +

→ n

n

n c

c , then the convergence radius of 



=0

2

n

n n xc is _______.

9. Find the convergence set of each series.

(1)  

= +1 2

1

2

n

n n

x n

(2)  

=

−+

1 3

)1(3

n

n

n

n

x

10. Find the sum function of the power series.

(1)  

=1n

n nx

(2)  

=

−

− −1

12

)11(, 12n

n

x n

x , and the sum of 



= −1 2)12(

1

n n

n .

11. Find the Taylor series of x

xf +

= 3

1 )( in )2( −x .

12. Recall that td t

x x

 −

= 0 2

1

1 arcsin ,

and find the first four nonzero terms in the Maclaurin series for xarcsin .