Chapter 7: Q2P (page 349)

(a) Prove that ${鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{0}^{蟺 / 2} c o s^{2} x d x$ by making the change of variable $x = \frac{1}{2} 蟺 - t$ in one of the integrals.
(b) Use the same method to prove that the averages of $s i n^{2} (n 蟺 x / l)$ and $c o s^{2} (n 蟺 x / l)$ are the same over a period.

Short Answer

Expert verified

(a).The solution is ${鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{0}^{蟺 / 2} c o s^{2} x d x$ .

(b). The solution is . ${鈭�}_{0}^{蟺濒濒 2} s i n^{2} \frac{n 蟺 x}{l} d x = {鈭�}_{0}^{蟺濒濒 2} c o s^{2} \frac{n 蟺 x}{l} d x$

Step by step solution

Given information

Prove that ${鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{0}^{蟺 / 2} c o s^{2} x d x$ by making the change of variable $x = \frac{1}{2} 蟺 - t$ in one of the integral.

Solve the integral from given information

a).

From the given information, ${鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{0}^{蟺 / 2} c o s^{2} x d x$ by making the change of variable $x = \frac{1}{2} 蟺 - t$ in one of the integrals is:

$d {鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{蟺 / 2}^{0} \sin^{2} (\frac{蟺}{2} - t) (- d t) = {鈭�}_{蟺 / 2}^{蟺} \cos^{2} t d t$

Thus, the solution is ${鈭�}_{0}^{蟺 / 2} s i n^{2} x d x = {鈭�}_{0}^{蟺 / 2} c o s^{2} x d x$ .

Obtain the averages of sin2(nπx/l) and cos2(nπx/l)

b).

From the given information, averages of $i n^{2} (n 蟺 x / l) a n d c o s^{2} (n 蟺 x / l)$ are the same over a period is given below.

The period is as follows:

$\frac{2 蟺}{T} = \frac{n 蟺}{l} 鈬� T = \frac{2 l}{n}$

Thus, $x = \frac{1}{2 n} - t .$ .

$d x = - d t$

Differentiate further as shown below.

{鈭�}_{0}^{蟺濒濒 2} s i n^{2} \frac{n 蟺 x}{l} = {鈭�}_{0 l 2 n}^{- 3 l 2 n} s i n^{2} (\frac{蟺}{2} - \frac{n 蟺}{l} t) d t (- 1) = {鈭�}_{- 3 l 2 n}^{1 l 2 n} \cos^{2} \frac{n 蟺 t}{l} d t = {鈭�}_{0}^{1 l 2 n} \cos^{2} \frac{n 蟺 t}{l} d t = {鈭�}_{0}^{1 l 2 n} \cos^{2} \frac{n 蟺 t}{l} d t

Thus, the solution is localid="1659241525185" ${鈭�}_{0}^{蟺濒濒 2} s i n^{2} \frac{n 蟺 x}{l} d x = {鈭�}_{0}^{蟺濒濒 2} c o s^{2} \frac{n 蟺 x}{l} d x$ .

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Short Answer

Step by step solution

Given information

Solve the integral from given information

Obtain the averages of sin2(nπx/l) and cos2(nπx/l)

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(a) Prove that 鈭�0蟺/2sin2xdx=鈭�0蟺/2cos2xdxby making the change of variablex=12蟺-t in one of the integrals.(b) Use the same method to prove that the averages ofsin2(n蟺x/l) andcos2(n蟺x/l) are the same over a period.

Short Answer

Step by step solution

Given information

Solve the integral from given information

Obtain the averages of sin2(nπx/l) and cos2(nπx/l)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Work Energy and Power

Space Physics

Measurements

Quantum Physics

Fields in Physics

Study anywhere. Anytime. Across all devices.