Chapter 7: Q2P (page 349)

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

Short Answer

Expert verified

(a).The solution is $\int_{0}^{π / 2} s i n^{2} x d x = \int_{0}^{π / 2} c o s^{2} x d x$ .

(b). The solution is . $\int_{0}^{πll 2} s i n^{2} \frac{n π x}{l} d x = \int_{0}^{πll 2} c o s^{2} \frac{n π x}{l} d x$

Step by step solution

Given information

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

Solve the integral from given information

a).

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

$d \int_{0}^{π / 2} s i n^{2} x d x = \int_{π / 2}^{0} \sin^{2} (\frac{π}{2} - t) (- d t) = \int_{π / 2}^{π} \cos^{2} t d t$

Thus, the solution is $\int_{0}^{π / 2} s i n^{2} x d x = \int_{0}^{π / 2} c o s^{2} x d x$ .

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

b).

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

The period is as follows:

$\frac{2 π}{T} = \frac{n π}{l} \Rightarrow T = \frac{2 l}{n}$

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

$d x = - d t$

Differentiate further as shown below.

\int_{0}^{πll 2} s i n^{2} \frac{n π x}{l} = \int_{0 l 2 n}^{- 3 l 2 n} s i n^{2} (\frac{π}{2} - \frac{n π}{l} t) d t (- 1) = \int_{- 3 l 2 n}^{1 l 2 n} \cos^{2} \frac{n π t}{l} d t = \int_{0}^{1 l 2 n} \cos^{2} \frac{n π t}{l} d t = \int_{0}^{1 l 2 n} \cos^{2} \frac{n π t}{l} d t

Thus, the solution is localid="1659241525185" $\int_{0}^{πll 2} s i n^{2} \frac{n π x}{l} d x = \int_{0}^{πll 2} c o s^{2} \frac{n π x}{l} d x$ .

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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 ∫0π/2sin2xdx=∫0π/2cos2xdxby making the change of variablex=12π-t in one of the integrals.(b) Use the same method to prove that the averages ofsin2(nπx/l) andcos2(nπx/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

Translational Dynamics

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