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For the polar functions r = 2Cos 20, and the radial lines, graph and calculate the area of the region R inside the curve of the function

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Answer to a math question For the polar functions r = 2Cos 20, and the radial lines, graph and calculate the area of the region R inside the curve of the function

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Fred
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68 Answers
Para graficar la función polar r = 2\cos(2\theta), primero necesitamos identificar el rango de \theta que queremos graficar. Dado que el periodo de \cos(2\theta) es \pi, podemos graficar un ciclo completo de la función tomando valores de \theta de 0 a \frac{\pi}{2}.

Para calcular el área de la región R dentro de la curva de la función, podemos usar la fórmula del área para curvas en coordenadas polares:

A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta

Donde r = 2\cos(2\theta) y \alpha = 0, \beta = \frac{\pi}{2}.

Paso 1: Graficar la función polar para \theta en el rango de 0 a \frac{\pi}{2}:
r = 2\cos(2\theta)

Paso 2: Calcular el área de la región R dentro de la curva de la función:
A = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} (2\cos(2\theta))^2 d\theta

A = \int_{0}^{\frac{\pi}{2}} 4\cos^2(2\theta) d\theta

Utilizando la identidad trigonométrica \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2}, tenemos:
A = \int_{0}^{\frac{\pi}{2}} 4\left(\frac{1 + \cos(4\theta)}{2}\right) d\theta

A = \int_{0}^{\frac{\pi}{2}} 2 + 2\cos(4\theta) d\theta

A = \left[2\theta + \frac{1}{2} \sin(4\theta)\right]_{0}^{\frac{\pi}{2}}

A = 2\cdot\frac{\pi}{2} + \frac{1}{2}\sin(2\pi) - 0 - 0

A = \pi

Entonces, el área de la región R dentro de la curva de la función es \pi.

\boxed{A = \pi}

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