Q. 67 Use a double integral with polar... [FREE SOLUTION]

Chapter 13: Q. 67 (page 991)

Use a double integral with polar coordinates to prove that the area of a sector with central angle $蠒$ in a circle of radius R is given by $A = \frac{1}{2} 蠒 R^{2} .$

Short Answer

Expert verified

The area of a sector with central angle $蠒$ is $A = \frac{1}{2} 蠒 R^{2}$

Step by step solution

Given information

The objective of this problem is to use double integral to prove that the area of a sector with central angle $蠒 i s \frac{1}{2} 蠒 R^{2} .$

calculation

In Cartesian system the equation of a circle vith radius R centered at origin is

$x^{2} + y^{2} = R^{2}$

Area of sector in double integration can be expressed as

$A = 鈭� d x d y$

In polar form

A=\int_{\phi}^{\phi} \int_{n}^{n} r d r d \theta $A = 鈭� d x d y$

Here, $蠒 = 0, 蠒_{2} = 蠒 a n d r_{1} = 0, r_{2} = R$

Then

$A = {鈭�}_{0}^{蠒} {鈭�}_{0}^{R} r d r d 胃 A = {鈭�}_{0}^{蠒} [{鈭�}_{0}^{R} r d r] d 胃 A = {鈭�}_{0}^{蠒} {[\frac{r^{2}}{2}]}_{0}^{R} d 胃$

$A = {鈭�}_{0}^{蠒} [\frac{R^{2} - 0}{2}] d 胃$

$A = \frac{1}{2} R^{2} {鈭�}_{0}^{蠒} d 胃$

$A = \frac{1}{2} R^{2} [胃]_{0}^{蠒}$

$A = \frac{1}{2} 蠒 R^{2}$

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91影视

Use a double integral with polar coordinates to prove that the area of a sector with central angle $蠒$ in a circle of radius R is given by $A = \frac{1}{2} 蠒 R^{2} .$

Short Answer

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Given information

calculation

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91影视

Use a double integral with polar coordinates to prove that the area of a sector with central angle 蠒 in a circle of radius R is given by A=12蠒R2.

Short Answer

Step by step solution

Given information

calculation

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Theoretical and Mathematical Physics

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Use a double integral with polar coordinates to prove that the area of a sector with central angle $蠒$ in a circle of radius R is given by $A = \frac{1}{2} 蠒 R^{2} .$