Apply the formula for the area of a region in polar coordinates. Determine a definite integral that represents the area of the region in the fourth quadrant enclosed by [tex]$r=5-\sin (\theta)$[/tex]. Enter your answer with limits of integration in [tex]$0 \leq \theta \leq 2 \pi$[/tex].
Answer (1)
To find the area of a region in polar coordinates, we use the formula for the area A of a region enclosed by a polar curve r = f ( θ ) :
A = 2 1 ∫ a b [ f ( θ ) ] 2 d θ
In this particular problem, the curve is given as r = 5 − sin ( θ ) and we want to find the area in the fourth quadrant.
The fourth quadrant corresponds to the interval for θ from 2 3 π to 2 π .
Thus, the definite integral that represents the area of the region enclosed by the curve in the fourth quadrant is:
A = 2 1 ∫ 2 3 π 2 π ( 5 − sin ( θ ) ) 2 d θ
This integral will provide the area of the specified region, which is formed by the portion of the curve r = 5 − sin ( θ ) in the fourth quadrant.
