How do you find the area bounded by the cardioid $r = 1 + cos (θ)$ $r=1+cos(theta)$ ?

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How do you find the area bounded by the cardioid $r = 1 + cos (θ)$ $r=1+cos(theta)$ ?

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Answer 1 · 2016-03-27T03:28:07

$A = \frac{(cos x + 4) sin x + 3 x}{4}$ $A=((cosx+4)sinx+3x)/4$
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Explanation:

Given: $r = 1 + cos (θ)$ $r=1+cos(theta)$
Required: Area of cardioid?
Solution Strategy: Polar Coordinate Area Integral
$A = \int_{θ_{1}}^{θ_{2}} \frac{1}{2} r^{2} d (θ)$ $A= int_(theta_1)^(theta_2) 1/2r^2d(theta)$ substitute for $r$ $r$
$A = \frac{1}{2} \int_{θ_{1}}^{θ_{2}} {(1 + cos (θ))}^{2} d (θ)$ $A= 1/2int_(theta_1)^(theta_2) ( 1+cos(theta))^2d(theta)$
$= \frac{1}{2} [\int (1 + 2 cos θ + {cos}^{2} θ) d (θ)]$ $=1/2[int (1+2costheta+cos^2theta) d(theta)]$
$= \frac{1}{2} [θ + 2 sin θ + \int {cos}^{2} θ d (θ)]$ $= 1/2[theta + 2sintheta+ int cos^2theta d(theta)]$
$I_{3} = \int {cos}^{2} θ d (θ)$ $I_3=color(brown)(int cos^2theta d(theta))$ apply reduction formula
$I_{3} = \frac{n - 1}{n} \int {cos}^{n - 2} θ d (θ) + \frac{{cos}^{n - 1} θ sin θ}{n}$ $I_3=(n-1)/n int cos^(n-2)thetad(theta) +(cos^(n-1)thetasintheta)/n$
$I_{3} = \frac{1}{2} θ + \frac{cos θ sin θ}{2}$ $I_3=color(brown)(1/2 theta +(costhetasintheta)/2)$
Putting it all together:

$A = \frac{(cos x + 4) sin x + 3 x}{4}$ $A=((cosx+4)sinx+3x)/4$

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How do you find the area bounded by the cardioid $r = 1 + cos (θ)$ $r=1+cos(theta)$ ?

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Explanation:

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Science

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How do you find the area bounded by the cardioid r=1+cos(θ)r=1+cos(theta)?

1 Answer

Explanation:

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How do you find the area bounded by the cardioid $r = 1 + cos (θ)$ $r=1+cos(theta)$ ?