Question

Calcule o comprimento da cardioide r=(1+cos 0) com a maio que 0

Answer 1

Resolução da questão, veja:

Para calcular o comprimento da cardioide, temos que:

$\mathsf{\dfrac{dr}{d\theta} = -sen(\theta)}}~\textsf{e}~\mathsf{L=\displaystyle\int\limits_{0}^{2\pi}~\sqrt{(-sen(\theta))^2+(1+cos(\theta))^2}}~\mathsf{d\theta}}$

Aplicando isso na questão dada, teremos:

$\mathsf{\dfrac{dr}{d\theta} = -sen(\theta)}}~\textsf{e}~\mathsf{L=\displaystyle\int\limits_{0}^{2\pi}~\sqrt{(-sen(\theta))^2+(1+cos(\theta))^2}}~\mathsf{d\theta}}\\ \\ \\ \mathsf{L=\displaystyle\int\limits_{0}^{2\pi}~\sqrt{sen^2(\theta)+(1+2cos(\theta)+cos^2(\theta))}}\ \mathsf{d\theta}}\\ \\ \\ \mathsf{L=\displaystyle\int\limits_{0}^{2\pi}~\sqrt{2+2cos(\theta)}}\ \mathsf{d\theta}}=\mathsf{\sqrt{2}~\cdot\displaystyle\int\limits_{0}^{2\pi}~\sqrt{1+cos(\theta)}}~\mathsf{d\theta}}$

Sabendo que:

$\mathsf{\dfrac{1+cos(\theta)}{2}}=\mathsf{cos^2\left(\dfrac{\theta}{2}\right)}$

Teremos então que:

$\mathsf{L=\sqrt{2}~\cdot\displaystyle\int\limits_{0}^{2\pi}~\sqrt{2cos^2\left(\dfrac{\theta}{2}}\right)}}}\ \mathsf{d\theta}=\mathsf{2~\cdot\displaystyle\int\limits_{0}^{2\pi}~\sqrt{cos^2\left(\dfrac{\theta}{2}}\right)}}\ \mathsf{d\theta}}$

Expressando a raiz em módulo, ficamos com:

$\mathsf{2~\cdot\displaystyle\int\limits_{0}^{2\pi}~\sqrt{cos^2\left(\dfrac{\theta}{2}}\right)}}\ \mathsf{d\theta}}=\mathsf{2~\cdot\displaystyle\int\limits_{0}^{2\pi}~\left|cos\left(\dfrac{\theta}{2}}\right)\right|}}\ \mathsf{d\theta}}$

Observando o sinal de cos(Θ/2), concluímos que:

$\mathsf{L=2~\cdot\displaystyle\int\limits_{0}^{\pi}~cos\left(\dfrac{\theta}{2}}\right)}}\ \mathsf{d\theta}} ~-\mathsf{2~\cdot\displaystyle\int\limits_{\pi}^{2\pi}~cos\left(\dfrac{\theta}{2}}\right)}}\ \mathsf{d\theta}}\\ \\ \\ \mathsf{L=4\left[sen\left(\dfrac{\theta}{2}\right)\right]_{0}^{\pi}} - \mathsf{4\left[sen\left(\dfrac{\theta}{2}\right)\right]_{\pi}^{2\pi}}\\ \\ \\ \Large\boxed{\boxedd{\boxed{\boxed{\boxed{\mathsf{L=8~u.c.}}}}}}}}}}}}}}}~\checkmark}}}}$

Ou seja, o comprimento dessa Cardioide é 8 Unidades de Comprimento.

Espero que te ajude. :-)

Bons estudos!