Polar coordinates：練習問題

A curve has polar equation \(r = 3\theta\) for \(0 \le \theta \le \pi\). Find the area of the region bounded by the curve and the line \(\theta = \pi\).

A curve has the polar equation \(r = a \cos 3\theta\) for \(-\frac{\pi}{6} \le \theta \le \frac{\pi}{6}\). Find the area of one loop of this curve.

Convert the Cartesian equation \((x^2 + y^2)^2 = a^2(x^2 - y^2)\) to polar form, where \(a > 0\).

A curve has polar equation \(r = 2 + \cos \theta\) for \(0 \le \theta < 2\pi\). Find the area of the region enclosed by the curve.

A curve has the polar equation \(r = 2a \sin^2 \theta\) for \(0 \le \theta \le \pi\), where \(a > 0\). Which of the following represents the area of the region enclosed by this curve?

A curve has the polar equation \(r = 2\sin \theta\) for \(0 \le \theta \le \pi\). Convert this equation into a Cartesian equation in terms of \(x\) and \(y\).

Find the area of the region enclosed by the loop of the polar curve \(r^2 = a^2 \cos 2\theta\).

Find the coordinates of the points on the polar curve \(r = 1 + \cos\theta\) for \(0 \le \theta < 2\pi\) where the tangent is perpendicular to the initial line.

A curve has the polar equation \( r = a(1 + \cos \theta) \) for \( 0 \le \theta \le \pi \), where \( a > 0 \).

(a) Find the area of the region enclosed by the curve and the initial line.
(b) Show that the distance of a point on the curve from the pole is maximized when \( \theta = 0 \) and find this maximum distance.
(c) Find the Cartesian equation of the curve in terms of \( x \) and \( y \).