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6.1 Exploring Accumulations of Change

Syllabus
2020
Topic
6.1
Level
—

Learning objectives

—CHA-4.A—Interpret the meaning of areas associated with the graph of a rate of change in context• CHA-4.A Interpret the meaning of areas associated with the graph of a rate of change in context.• CHA-4.A.1 The area of the region between the graph of a rate of change function and the x axis gives the accumulation of change.• CHA-4.A.2 In some cases, accumulation of change can be evaluated by using geometry.• CHA-4.A.3 If a rate of change is positive (negative) over an interval, then the accumulated change is positive (negative).• CHA-4.A.4 The unit for the area of a region defined by rate of change is the unit for the rate of change multiplied by the unit for the independent variable.• Enduring understanding CHA-4: Definite integrals allow us to solve problems involving the accumulation of change over an interval.

Read Accumulated Change from a Rate Graph

The signed area between a rate graph and the horizontal axis over an interval is the accumulated change in the original quantity. Area above the axis contributes positively; area below the axis contributes negatively.

\text{accumulated-change units}=(\text{rate units})(\text{input units})

Split the region wherever the graph crosses the axis or changes geometric shape. Find each rectangle, triangle, or other familiar area, attach a positive or negative sign according to its position relative to the axis, and add the signed contributions. If an initial amount is given, then final amount = initial amount + net accumulated change.

Suppose a tank's net flow-rate graph, in liters per minute, forms a triangle above the axis with base 44 minutes and height 66 liters per minute, followed by a triangle below the axis with base 22 minutes and height 33 liters per minute. The net change is 12(4)(6)−12(2)(3)=12−3=9\tfrac12(4)(6)-\tfrac12(2)(3)=12-3=9 liters. If the tank began with 2020 liters, it ends with 2929 liters.

Net change is not the same as total geometric area or final amount. In the example, the total unsigned area is 12+3=1512+3=15 liters, the net change is 99 liters, and the final amount is 2929 liters. Preserve the sign before interpreting the context.