Chapter 9.1, Problem 3CP

(a) Using calculus, find the area bounded by the two parabolas $P_1(x)=x^2-x+1/2$

and $P_2(x)=-x^2+x+1/2$ . (b) Estimate the area as a Type I Monte Carlo simulation, by finding the average value of $P_2(x)-P_1(x)$

on [0, 1]. Find estimates for $n={10}^i$ for $2\le i\le 6$ . (c) Same as (b), but estimate as a Type 2 Monte Carlo problem: Find the proportion of points in the square [0. 1] x [0. 1) that lie between the parabolas. Compare the efficiency of the two Monte Carlo approaches.