## Plane Areas in Rectangular Coordinates

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

1. by using a horizontal element (called strip) of area, and
2. 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...

$\displaystyle A = {\int_{y_1}^{y_2}} x \, dy = {\int_{y_1}^{y_2}} (x_R - x_L) \, dy$

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...

$\displaystyle A = {\int_{x_1}^{x_2}} y \, dx = {\int_{x_1}}^{x_2} (y_U - y_L) \, dx$

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...

$\displaystyle A = \frac{1}{2}{\int_{\theta_1}^{\theta_2}} r^2 \, d\theta$

Where θ1 and θ2 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 dθ. Note that r is a polar function or r = f(θ). See figure above.

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