Chapter 8.3, Problem 75E

a.

To determine

To find: The area of the region under the curves $f_1\left(x\right)={\sin }^{2}x$ and $f_2\left(x\right)={\sin }^{2}2x$ on the interval $\left[0,\pi \right]$.

b.

To determine

To find: The area of the region bounded by the graphs of $f_n\left(x\right)={\sin }^{2}nx$ on the interval $\left[0,\pi \right]$ where n is positive integer and interpret the result.

c.

To determine

To prove: The integral ${\displaystyle {\int }_0^{\pi }{\sin }^2\left(nx\right)}dx$ has the same value where n positive integer.

d.

To determine

To check: Whether the result obtained part (c) exists, if sine is replaced by cosine in the integral ${\displaystyle {\int }_0^{\pi }{\sin }^2\left(nx\right)}d$.

e.

To determine

To find: The integral ${\displaystyle {\int }_0^{\pi }{\sin }^{4}x}dx$ using part (a) to part (c) and interpret the result.