# 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 − x² 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 = x² − 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 x² + y² = 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.