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The following properties can be fairly easily deduced from the Riemann sum approach to integration:
 Additive properties:
 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
20050114