Test 2 review
-------------

PART I
------
1. Algebraically determine whether the given integral is convergent
   or divergent, if it converges find the exact limit:

    int_0^1 1/x^(2/3)dx
   
    WebW. 6.6, 12, 13, 14

2. Find the exact area of the region bounded by the curves ...

   WebW. 7.1, 2

     y=x^2-1, y=1-x
To find the intersection solve

x^2-1 = 1 - x
x^2+x-1=0

plot({x^2-1,1-x},0.01,x=-2.1..1.1)

 Int(2-x-x^2,x={simpson,0.001,-2..1})=4.5

roots({2,-1,-1})={1,0;-2,0}

roots({2,-1,-1})
p(x) = 2.0 + -1.0*X^1 + -1.0*X^2
{1,0}
{-2,0}
Real roots: 1,-2

3. Which expression represents the area of the region...

4. Using the Method of Disks/Washers, find the volume of the solid
   obtained by revolving the region bounded by the curves ... 
  about the line y=a.

    WebW. 7.3, 2, 3

Problem 3

plot({6*sqrt(x+0.001),6*x^2},0.01,x=0.01..1.1)

5. Using the method of disks/washers, which expression 
   represents the volume of the solid obtained by rotating 
  the region bounded by ... about the y-xis?


6. Using the method of cylindrical shells, which expression 
   represents the volume of the solid obtained by rotating 
   the region bounded by ... about the y-axis?

7. Determine whether the sequence converges or diverges and if it converges find the limit; 
     
         an = r^n

       WebW. 8.1, 3

8. Determine whether the series converges or diverges and if it converges find the sum.

     WebW. 8.2, 5, 6

9. Determine whether the series converges or diverges and if it converges find the sum.

     WebW. 8.2, 7, 8, 9, 10


10. Describe the behavior of the geometric series ...

  WebW. 8.2, 3, 6
   

   
  PART II  (Free response )
  -------------------------

  11.  Algebraically find the exact arc length of the curve
      
   WebW. 7.4, 3, 5-8

 12.
      A tank is full of water. Its shape is obtained by rotating the curve y = x^1.5, 0<=x<=2 around the y-axis.
       Given the density of water is 1000 kg/m^3 
      calculate work required to pump all the water out of the tank. Graviational constant is 9.8 m/s^2.

      WebW. 7.6, 5, 6

      
 13.  Calculate an improper integral.
       
       WebW. 6.6, 5

 14. If X J of work are needed to stretch a spring from "a" cm to "b" cm and Y J are needed 
       to stretch it from "c" cm to "d" cm, what is the natural length of the spring?