Chapter 7.6, Problem 52E

Graphical and Numerical Reasoning Consider the region bounded by the graphs of $y=x^{2n}$ and $y=b$, where $b>0$ and n is a positive integer.

(a) Sketch a graph of the region.

(b) Set up the integral for finding $M_y$. Because of the form of the integrand, the value of the integral can be obtained without integrating. What is the form of the integrand? What is the value of the integral and what is the value of $\bar{x}?$

(c) Use the graph in part (a) to determine whether $\bar{y}>\frac{b}{2}$ or $\bar{y}<\frac{b}{2}$- Explain.

(d) Use integration to find y as a function of n.

(e) Use the result of part (d) to complete the table.

n	1	2	3	4
$\bar{y}$

(f) Hind $\underset{n\to \infty }{\lim }\bar{y}$

(g) Give a geometric explanation of the result in part (f).