Q12.2P Use a graph of sin2胃聽and the t... [FREE SOLUTION]

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Chapter 11: Q12.2P (page 558)

Use a graph of $s i n^{2} 胃$ and the text discussion just before (12.4)to verify the equations (12.4). Note that the area under the $s i n^{2} 胃$ graph from $0 - \frac{蟺}{2}$ and the area from $\frac{蟺}{2} - 蟺$ are mirror images of each other, and this will be true also for any function of $\sin^{2} 胃$ .

Short Answer

Expert verified

The statement is verified.

Step by step solution

Given Information

The function is $\sin^{2} 胃$ .

Definition of elliptic form.

The elliptic form of the integral is defined as $E (\frac{蟺}{0}, k) = {鈭�}_{0}^{\frac{蟺}{2}} \sqrt{1 - k^{2} s i n^{2} 胃} d 胃$ .

Verify the statement.

The function is $\sin^{2} 胃$ .

$\begin{array}{rcl} l (n 蟺卤蠒) & = & {鈭�}_{0}^{n 蟺卤蠒} f (\sin^{2} 胃) d 胃 (0 鈮� 鈭� 鈮� \frac{蟺}{2}) \\ = & {鈭�}_{0}^{n 蟺} f (\sin^{2} 胃) d 胃 + {鈭�}_{n 蟺}^{n 蟺卤蠒} f (\sin^{2} 胃) d 胃 \\ = & n {鈭�}_{0}^{蟺} f (\sin^{2} 胃) d 胃 + {鈭�}_{0}^{蠒} f (\sin^{2} 胃) d 胃 \end{array}$

Find the value of $\begin{array}{rcl} n {鈭�}_{0}^{蟺} f (\sin^{2} 胃) d 胃 \end{array}$

$\begin{array}{rcl} {鈭�}_{0}^{蟺} f (\sin^{2} 胃) d 胃 & = & {鈭�}_{0}^{\frac{蟺}{2}} f (\sin^{2} 胃) d 胃 + {鈭�}_{\frac{蟺}{2}}^{蟺} f (\sin^{2} 胃) d 胃 \\ = & {鈭�}_{\frac{蟺}{2}}^{0} f (\sin^{2} (蟺 - 胃)) (- d 胃) \\ = & {鈭�}_{0}^{\frac{蟺}{2}} f (\sin^{2} 胃) d 胃 \\ = & 2 {鈭�}_{0}^{\frac{蟺}{2}} f (\sin^{2} 胃) d 胃 \end{array}$

Hence, $\begin{array}{rcl} l (n 蟺卤蠒) & = & 2 n I n (\frac{蟺}{2}) 卤 l (蠒) \end{array}$ .

The statement is verified.

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Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Verify the statement.

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Use a graph of sin2胃and the text discussion just before (12.4)to verify the equations (12.4). Note that the area under thesin2胃 graph from 0-蟺2and the area from 蟺2-蟺 are mirror images of each other, and this will be true also for any function ofsin2胃.

Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Verify the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Electromagnetism

Mechanics and Materials

Linear Momentum

Magnetism and Electromagnetic Induction

Electrostatics

Study anywhere. Anytime. Across all devices.