6.1 Exploring Accumulations of Change

Accumulating a rate

A definite integral accumulates a rate of change over an interval to give the net change in a quantity. The area under a rate curve equals the accumulated amount.

Examples

Integrating velocity over time gives displacement; integrating a flow rate gives the total volume that flowed. The integral turns a rate back into a total.

Key terms

Accumulation

A definite integral accumulates a rate into a net change.

Area under a rate

Equals the accumulated quantity.

Velocity → displacement

Integrating velocity gives displacement.

Flow rate → volume

Integrating a flow rate gives total volume.

Big idea

Integration turns a rate back into a total.

Try an AP-style question

Water flows into a tank at a constant 5 gallons per minute for 8 minutes. How much water enters?

1. 13 gallons
2. 5/8 gallon
3. 8 gallons
4. 40 gallons