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:

Area of a Positive Function

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.

x-intercepts

Parabola

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

Area of a Function Operations

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

Area of a Function·

Area of a Function Solution

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:

Equation of a Line

The equation of the straight line that passes through BC:

Equation of a Line

Triangle

Integrals

Integrals

Integrals Solution


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:

Area of a Negative Function

Examples

1. Calculate the area of the site bounded by the curve y = x2 − 4x and x-axis.

Intercepts

Negative Function

Integrals

Integrals Solution


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

Area of a Function

Integrals

Integrals Solution

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 Equation of a Line, x = 0, x = 4 and x-axis.

Area of a Function

Integrals

Integrals

Integrals Solution

2.Calculate the area of the plane region bounded by the circle x2 + y2 = 9.

Area of a Quadrant

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

Indefinite Integral

Indefinite Integral

Indefinite Integral

Integral Operations

Integral Operations

Find the new limits of integration.

Change of Variable

Change of Variable

Integrals

Area of Circular Section