Due:  Wednesday, March 12

Part I:   Using a Monte Carlo method, estimate the area between e^x and the polynomial 1 + x + x²/2 over the interval 0 < x < 2. .  Be certain to draw a graph of the region.   Check your answer by actual integration.

Part II:   Suppose that a cylindrical hole of radius 1 cm is drilled through a sphere of radius 2 cm.  (Assume that the hole is drilled so that its axis is also an axis of the sphere.)  Using a Monte Carlo method, estimate the amount (in cubic cm) of the sphere that is removed by the hole.   Hints:  Assume that the sphere is centered at the origin and that the hole is perpendicular to the xy-plane.  Then a random point P = (x,y,z) is in the hole if the distance from (x,y) to (0,0) is less than 1 and the distance from P to (0,0,0) is less than 2.
 

Part III:   Let us estimate the number of prime numbers between 1 and 10¹².   Generate a large number (at least 500) random 12 digit integers.  Using an appropriate loop and conditional statement, compute the percentage of these numbers that are prime.  Try doubling or tripling your original number of integers to obtain more precision.  Be careful however, since at some point you will need to wait a large amount of time before Maple is finished with the computations.

Part IV:   Discuss briefly the history of Monte Carlo methods.

Part V:   Look up the statement of the Prime Number Theorem.  According to the Prime Number Theorem, approximately what percentage of the numbers between 1 and 10¹² are prime?  Explain your reasoning.