Applications of integration

Represent area, distance, and volume as accumulation and check signs and intervals.

Meaning

Integration accumulates a rate to find a total change or stacks cross-sectional areas to find a volume. The meaning and units depend on which quantity is accumulated with respect to which variable.

Example

If water flows in at L per minute for the amount entering is L. This accumulates amounts found by multiplying a rate by a time width.An antiderivative of 2t is t², so the accumulation is [t²]₀³=9−0=9 L. The inflow rate is nonnegative on this interval, so signed change equals total inflow.Integrating velocity gives displacement. Distance traveled is found by integrating speed, the absolute value of velocity. Distinguish signed total change from total distance.

Practice for this concept

Scope and references

Topics

Areas between curves and axes or between two curves, distance traveled, volume

Learning goal

Represent area, distance, and volume as accumulation and check signs and intervals.

Helpful background

Check the interval, signs, units, and accumulated quantity.

Applications of integrals

These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.

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