Chapter 13: Q. 67 (page 991)

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

Short Answer

Expert verified

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

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 $ϕ i s \frac{1}{2} ϕ R^{2} .$

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

$A = \iint d x d y$

In polar form

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

Here, $ϕ = 0, ϕ_{2} = ϕ a n d r_{1} = 0, r_{2} = R$

Then

$A = \int_{0}^{ϕ} \int_{0}^{R} r d r d θ A = \int_{0}^{ϕ} [\int_{0}^{R} r d r] d θ A = \int_{0}^{ϕ} {[\frac{r^{2}}{2}]}_{0}^{R} d θ$

$A = \int_{0}^{ϕ} [\frac{R^{2} - 0}{2}] d θ$

$A = \frac{1}{2} R^{2} \int_{0}^{ϕ} d θ$

$A = \frac{1}{2} R^{2} [θ]_{0}^{ϕ}$

$A = \frac{1}{2} ϕ R^{2}$

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

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