Name:_________________________
Q
P
1
2
/6
3
/3
/6
Total
/15
MATH 1312 Calculus II
Fall 2006 Test II
Thursday, October 19, 2006
Instruction: Please show all work! Full credit for a problem can only be given when
the answers are derived with the correct steps.
Question 1 [6 points]
Consider the region bounded by the curve y  x 2 , the horizontal line y  1, and the y-axis.
Find the volume of the solid obtained by rotating the region around the
(a) x-axis.
(b) y-axis.
(c) The line y  1 .
Question 2 [3 points]
For a  b, the arc length of the curve y  f ( x) from x  a to x  b is given by
Arc Length  
b
a
1  ( f '( x)) 2 dx
(a) Use the arc length formula above to find the length of the curve f ( x)  1  x 2 from
x  0 to x  1 .
(b) Without using the above formula, can you arrive at the same answer? Explain how.
Question 3 [6 points]
The density of oil in a circular oil slick on the surface of the ocean at a distance r meters
from the center of the slick is given by  (r )  10 /(1  r ) kg/m 2 .
(a) If the slick extends from r  0 to r  5000 m, write down a Riemann sum
approximating the total mass of oil in the slick.
(b) Find the exact value of the mass of oil in the slick by turning your sum into an
integral and evaluating it.
(c) Within what distance r is half the oil of the slick contained?
*** End of Test ***