The graph of the polar curve $r = 1 - 2 \cos \theta$ for $0 \leq \theta \leq \pi$ is shown above. Let $S$ be the shaded region in the third quadrant bounded by the curve and the $x$-axis. (a) Write an integral expression for the area of $S$. (b) Write expressions for $\frac { d x } { d \theta }$ and $\frac { d y } { d \theta }$ in terms of $\theta$. (c) Write an equation in terms of $x$ and $y$ for the line tangent to the graph of the polar curve at the point where $\theta = \frac { \pi } { 2 }$. Show the computations that lead to your answer.
The graph of the polar curve $r = 1 - 2 \cos \theta$ for $0 \leq \theta \leq \pi$ is shown above. Let $S$ be the shaded region in the third quadrant bounded by the curve and the $x$-axis.
(a) Write an integral expression for the area of $S$.
(b) Write expressions for $\frac { d x } { d \theta }$ and $\frac { d y } { d \theta }$ in terms of $\theta$.
(c) Write an equation in terms of $x$ and $y$ for the line tangent to the graph of the polar curve at the point where $\theta = \frac { \pi } { 2 }$. Show the computations that lead to your answer.