Properties of Definite Integral II

Definite integrals are essential tools in calculus, used to quantify accumulated change over an interval. A common physical application is calculating the total displacement from a velocity-time graph. If a velocity function, v(t), describes the motion of an object over time, the definite integral gives the net displacement between times a and b. This integral corresponds to the signed area under the velocity curve between those two points.

Two fundamental properties of definite integrals aid in simplifying such calculations: the additivity property and the constant multiple property. The additivity property states that if a continuous function is integrated over adjacent intervals, the sum of these integrals equals the integral over the combined interval. Mathematically, for a < c < b,

This is particularly useful when analyzing motion over segmented intervals, such as a car’s journey divided into different phases. It allows one to compute displacement piecewise and sum the results.

The constant multiple property expresses that scaling a function by a constant scales the integral by the same factor. For a constant, c,

This property is especially relevant when considering uniform changes in velocity—such as doubling the speed throughout the trip—which results in a proportional increase in total displacement. Together, these properties streamline the application of definite integrals in practical scenarios, such as kinematic analysis.