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 setup to finding $M_y$, and the value of the integral along with the value of $\overline{x}$ where, the equations of graph are $y=x^{2n}$ and $y=b$, where $b>0$ and n is positive.

(c)

To determine

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

(d)

To determine

To calculate: The value of $\overline{y}$ using integration where, the equations of graph are $y=x^{2n}$ and $y=b$, where $b>0$ and n is positive.

(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 $\underset{n\to \infty }{\lim }\overline{y}$ where, the equations of graph are $y=x^{2n}$ and $y=b$, where $b>0$ and n is positive.

(g)

To determine

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