Question

Find the area of the region that lies inside the first curve and outside the second curve.

r = 3 cos θ,      r = 1 + cos θ

Answer 1

Step 1:

The curves are and .

Find the intersection points by equating the two curves.

The area of curve .

The intersection points are the limits of the integration.

The area lies between to .

Required area is inside the first curve and out side the second curve.

So, the diffrernce between first curve and second curve.

Step 2:

Graph the curves and .

Observe the graph:

The curves are symmetric about horizontal axis , so the area,

Answer 2

Contd...

Step 3:

Apply sum and difference formula in integration .

The area square units.

Solution:

The area is square units.