# Area of a Function

## Area between a Function and the x-axis

### 1. The Function is Positive

If the function is positive on an interval [a, b] when the graph of the function is above the x-axis. The **area function** can be defined by:

To find the area, follow these steps:

1. Calculate the **x-intercepts** by making f(x) = 0 and solving the equation.

2.The **area** is equal to the **definite integral of the function** whose integration limits are the **x-intercepts**.

#### Examples

1.Calculate the area of the site bounded by the curve y = 9 − x^{2} and the x-axis.

First, find the x-intercepts to the curve and determine the limits of integration.

Since the parabola is symmetrical about the y-axis, the area is twice the area between x = 0 and x = 3.

2.Determine the area of the region enclosed by the function xy = 36, the lines x = 6 and x = 12 and the x-axis.

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3.Calculate the area of the triangle that is formed by joining the points A(3, 0), B(6, 3) and C(8, 0).

The equation of the straight line that passes through AB:

The equation of the straight line that passes through BC:

### 2. The Function is Negative

If the function is negative in an interval [a, b] then the graph of the function is below the horizontal axis. The area of the function can be defined by:

#### Examples

1. Calculate the area of the site bounded by the curve y = x^{2} − 4x and x-axis.

2. Find the area bounded by the curve y = cos x and the x-axis between π/2 and 3π/2.

### 3. The Function Has Positive and Negative Values

In this case, the enclosure has areas above and below the x-axis. To calculate the area of the function follow these steps:

1. Calculate the **x-intercepts** by making f(x) = 0 and solve the equation.

2.Order the roots from smallest to largest, which are the limits of integration.

3.The **area** is equal to the** sum of the definite integral **in the absolute value of each interval.

#### Examples

1.Find the area bounded by the lines , x = 0, x = 4 and x-axis.

2.Calculate the area of the plane region bounded by the circle x^{2} + y^{2} = 9.

The area of the circle is four times the area enclosed by the first quadrant and the coordinate axes.

Find the new limits of integration.