Properties of Definite Integral II

Definite integrals measure accumulation, such as total displacement from a velocity-time curve. Certain properties simplify these calculations, including additivity and the constant multiple properties.

For example, a car traveling at varying speeds has its total displacement shown by the area under the velocity-time curve.

When the trip is divided into segments, the additivity property of definite integrals becomes useful.

Suppose the car travels from a to c, and then from c to b, with a continuous velocity function.

The total displacement from a to b is found by integrating the velocity from a to c, and then adding the integral from c to b.

This shows that the definite integral from a to b equals the sum of the integrals over the adjacent intervals.

Another key property is the constant multiple property. This applies when the car’s velocity is scaled.

When the original velocity is doubled for the whole trip, the total displacement—the area under the curve—is also doubled.

This means the integral of a constant times a function equals the constant multiplied by the integral of that function.