8. Consider the polar curve given by the equation $$ r=2 \sin ( heta) $$ a) (7pts) The point $$ \left(\sqrt{2}, \frac{\pi}{4} ight) $$ is a point on this polar curve. Determine the $$ x $$ )-coordinates of this point. b) (8pts) Represent this curve as a parametric curve (with parameter $$ heta $$ ). c) (8pts) Set up, but do not evaluate, an integral which represents the area enclosed by this curve.

Question

8. Consider the polar curve given by the equation $$ r=2 \sin (	heta) $$ a) (7pts) The point $$ \left(\sqrt{2}, \frac{\pi}{4}ight) $$ is a point on this polar curve. Determine the $$ x $$ )-coordinates of this point. b) (8pts) Represent this curve as a parametric curve (with parameter $$ 	heta $$ ). c) (8pts) Set up, but do not evaluate, an integral which represents the area enclosed by this curve.

Huang Cross · Accepted Answer

a) The $$ x $$-coordinate of the point $$ \left(\sqrt{2}, \frac{\pi}{4}\right) $$ is $$ \boxed{1} $$. b) The parametric representation of the curve is: $$ \begin{cases} x(\theta) = \sin(2\theta) \ y(\theta) = 2 \sin^2(\theta) \end{cases} $$ c) The integral representing the area enclosed by the curve is: $$ \boxed{2 \int_0^{\pi} \sin^2(\theta) \, d\theta} $$

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