12.7: Numerical Integration

Integration is often used to calculate the area under a curve. In engineering, the area under a curve can also represent accumulated quantities. For example, integrating velocity over time gives displacement, and integrating flow rate over time gives volume.

\(
\int_a^b f(x)\,dx
\)

Approximating Area with trapz

The trapz function estimates area using the trapezoidal rule. The basic idea is to divide the area under a curve into trapezoids and add their areas.

The basic format is:

area = trapz(x, y);

For a quick example, consider the function \(f(x) = 3*x^2 - 1 \) from \(x = 2\) to \(x = 4\).

f = @(x) 3*x.^2 - 1;
x = [2 4];
y = f(x);
area = trapz(x, y)

Using only two x-values creates one large trapezoid, which may not be very accurate for a curved function. A better approach is to use more points.

Example \(\PageIndex{1}\)

Calculating the integral of a function using trapz.

f = @(x) 3*x.^2 - 1;
x = linspace(2, 4, 50);
y = f(x);
area = trapz(x, y)

Solution

area = 54.002

Note

Key Idea

The more points you use with trapz, the smaller the trapezoids become. Smaller trapezoids usually give a better approximation for smooth curves.

More Accurate Integration with integral

MATLAB also provides the integral function, which numerically integrates a function over an interval. Instead of giving x and y data points, we give MATLAB the function handle and the limits of integration.

The basic format is:

area = integral(functionHandle, xmin, xmax);

Example \(\PageIndex{1}\)

Calculating the integral of a function using integral.

f = @(x) 3*x.^2 - 1;

area = integral(f, 2, 4)

Solution

area = 54

For many smooth functions, integral is more accurate and easier to use than trapz because MATLAB controls the numerical approximation internally.

Function	Inputs	Best Used When
trapz	x data and y data	You already have sampled data points
integral	function handle and limits	You have a mathematical function