## 07 Area Enclosed by r = 2a cos θ and r = 2a sin θ

**Problem**

Find the area enclosed by the following:

(b) $r = 2a \sin \theta$

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## 05 Area Enclosed by r = a sin 2θ and r = a cos 2θ

**Problem**

Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.

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**Example 6**

What is the area within the curve *r*^{2} = 16 cos θ?

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## 02 Area Bounded by the Lemniscate of Bernoulli r^2 = a^2 cos 2θ

**Example 2**

Find the area bounded by the lemniscate of Bernoulli r^{2} = a^{2} cos 2θ.

**Problem**

Find the area enclosed by

• *r* = 2*a* sin^{2} θ

• *r* = 2*a* cos^{2} θ

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## Example 8 | Area bounded by arcs of quarter circles

**Problem**

Arcs of quarter circles are drawn inside the square. The center of each circle is at each corner of the square. If the radius of each arc is equal to 20 cm and the sides of the square are also 20 cm. Find the area common to the four circular quadrants. See figure below.

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## Example 7 | Area inside the square not common to the quarter circles

**Problem**

The figure shown below is composed of arc of circles with centers at each corner of the square 20 cm by 20 cm. Find the area inside the square but outside the region commonly bounded by the quarter circles. The required area is shaded as shown in the figure below.

## Example 6 | Plane Areas in Rectangular Coordinates

**Example 6**

Find each of the two areas bounded by the curves *y* = *x*^{3} - 4*x* and *y* = *x*^{2} + 2*x*.

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## Example 3 | Plane Areas in Rectangular Coordinates

**Example 3**

Find the area bounded by the curve *x* = *y*^{2} + 2*y* and the line *x* = 3.

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