   Next: The mean value theorem Up: Integrated Calculus II Spring Previous: Error estimates

## Some properties of integrals

The following properties can be fairly easily deduced from the Riemann sum approach to integration:   • Scaling by a constant: • Integral of a sum: • Area interpretation of the integral:  Here we have:
• is the area bounded by the -axis, the lines and and the part of the graph of , where .
• is the area bounded by the -axis, the lines and and the part of the graph where .
• Distance interpretation of the integral.

Suppose that is the velocity at time of a particle moving along the -axis (note that this can be of any sign).  Here we have:
• is the total distance traveled in the forward direction.
• is the total distance traveled in the backward direction.
• is the total distance traveled (relevant, for example, for your gasoline expenditure).
• is the net displacement from the initial position (relevant, for example, for knowing where you end up).
• Integral inequalities:
• If and , then .
• If and , then .   Next: The mean value theorem Up: Integrated Calculus II Spring Previous: Error estimates
George A. J. Sparling 2005-01-14