## Plane Areas in Rectangular Coordinates

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

From the figure, the area of the strip is $x \, dy$, where $x = x_R - x_L$. The total area can be found by running this strip starting from $y_1$ going to $y_2$. Our formula for integration is...

Note that $x_R$ is the right end of the strip and is always on the curve $f(y)$ and $x_L$ is the left end of the strip and is always on the curve $g(y)$. We therefore substitute $x_R = f(y)$ and $x_L = g(y)$ prior to integration.

### Using Vertical Strip

We apply the same principle of using horizontal strip to the vertical strip. Consider the figure below.

The total area is...

Where

$y_U$ = upper end of the strip = $f(x)$

$y_L$ = lower end of the strip = $g(x)$

The steps in finding the area can be outlined as follows:

1. Sketch the curve

2. Decide what strip to use and define its limits

3. Apply the appropriate formula based on the strip then integrate.

## Plane Areas in Polar Coordinates

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

Where θ_{1} and θ_{2} are the angles made by the bounding radii.

The formula above is based on a sector of a circle with radius *r* and central angle *d*θ. Note that *r* is a polar function or *r* = *f*(θ). See figure above.