How do we recover the total change in a quantity from its rate of change?
Use the definite integral of a rate of change to find the total or net change in a quantity over an interval
WACE Year 12 Mathematics Methods Unit 4 total change from a rate: integrating a rate of change to find net change, recovering a quantity from its rate plus an initial value, with worked SCSA-style examples in real contexts.
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What this dot point is asking
SCSA Unit 4 applies the definite integral to accumulation: recovering total change from a rate. This is the Fundamental Theorem of Calculus read in context, and this dot point asks you to interpret and compute it in real situations such as flow, growth and consumption. It is examined in both sections.
The accumulation principle
The Fundamental Theorem of Calculus says that integrating a rate of change gives the net change in the original quantity.
The units confirm the interpretation: a rate in litres per minute integrated over minutes gives litres. A positive integral means a net increase; a negative integral means a net decrease.
Recovering the quantity itself
If the initial value of the quantity is known, add it to the accumulated change to find the quantity at a later time.
Net versus total accumulation
When the rate changes sign, the definite integral gives the net change, with inflow and outflow cancelling. If a question asks for the total amount that flowed in (ignoring outflow), split the integral where the rate is zero and add the absolute values of the positive and negative parts, exactly as for distance travelled in kinematics.