Properties of Definite Integral I

A car travels with a velocity that changes over time. Its total displacement is found using a definite integral—the area under the velocity-time curve. Several properties simplify this accumulation, like linearity and integrating constant functions.

The first property is linearity, which applies to the addition and subtraction of functions.

Consider two different trips, each with a parabolic velocity-time curve.

Adding the velocity functions of these trips creates a new combined velocity curve.

The area under the combined curve gives the total displacement by adding the integrals of the two velocity functions.

This shows that the integral of a sum equals the sum of the individual integrals.

Subtracting the two velocity functions creates a curve showing the velocity difference. The area under this difference curve gives the difference in displacement between the trips.

This shows that the integral of the difference equals the difference of the individual integrals.

The second property is integrating constant functions. When velocity stays constant, the velocity-time curve becomes a horizontal line.

The area under the line gives total displacement, showing that integrating a constant equals the constant multiplied by the time interval.