Question

Use a graphing utility to graph the polar equation and find the area of the given region.

Inner loop of r = 1 + 2 sin θ

Answer 1

Step 1:

The polar curve is .

Graph:

Graph the polar curve . 

 

Observe the graph, the  region shaded in blue color is the required inner loop area.

Limits of the the inner circle can be found by substituting .

is negative in third and fourth quadrant.

In third quadrant:

In fourth quadrant:

 Thus, the limits of integral are and .

Step 2:

Area of the curve in polar form is .

Area of the one loop of polar curve is .

Solution:

Area of the one loop of polar curve is .