# Area by Integration

## Example 4 | Plane Areas in Rectangular Coordinates

**Example 4**

Solve the area bounded by the curve *y* = 4*x* - *x*^{2} and the lines *x* = -2 and *y* = 4.

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

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## 01 Area Enclosed by r = 2a sin^2 θ

**Example 1**

Find the area enclosed by *r* = 2*a* sin^{2} θ.

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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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## Example 2 | Plane Areas in Rectangular Coordinates

**Example 2**

Find the area bounded by the curve *a*^{2}*y* = *x*^{3}, the *x*-axis and the line *x* = 2*a*.

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## Plane Areas in Polar Coordinates | Applications of Integration

The fundamental equation for finding the area enclosed by a curve whose equation is in polar coordinates is...

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## Plane Areas in Rectangular Coordinates | Applications of Integration

There are two methods for finding the area bounded by curves in rectangular coordinates. These are...

- by using a horizontal element (called strip) of area, and
- by using a vertical strip of area.

The strip is in the form of a rectangle with area equal to length × width, with width equal to the **differential element**. To find the total area enclosed by specified curves, it is necessary to sum up a series of rectangles defined by the strip.

### Using Horizontal Strip

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