Chapter 7.6, Problem 52E

(a)

To determine

To graph: The region bounded by the graphs of $y=x^{2n}$ and $y=b$, where $b>0$ and n is positive.

(b)

To determine

To calculate: The form of integrand, the integral for finding $M_y$ and the value of the integral as well as the value of $\overline{x}$ for the given equations of graphs.

(c)

To determine

Whether $\overline{y}>\frac{b}{2}\text{ or }\overline{y}<\frac{b}{2}$, using the region bounded by the given graphs.

(d)

To determine

To calculate: The value of $\overline{y}$ using integration for given equations of the graphs.

(e)

To determine

To fill: The blank spaces in the following table:

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

Using the value of $\overline{y}$ from the part (d)

$\frac{2n+1}{4n+1}b$

(f)

To determine

To calculate: The value of $\underset{n\to \infty }{\lim }\overline{y}$ for given equation of graphs.

(g)

To determine

A geometric explanation for result $\overline{y}=\frac{1}{2}b$.