How does the Fundamental Theorem of Calculus link differentiation and integration into inverse processes?
State and apply the Fundamental Theorem of Calculus to evaluate definite integrals and to differentiate integral functions
WACE Year 12 Mathematics Methods Unit 4 the Fundamental Theorem of Calculus: both forms linking differentiation and integration, evaluating definite integrals as F(b) minus F(a), differentiating an integral function, with worked SCSA-style examples.
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What this dot point is asking
SCSA Unit 4 emphasises the Fundamental Theorem of Calculus as the bridge between the definite integral (a limit of sums) and antidifferentiation. This dot point asks you to state and apply both forms, examined in both sections of the WACE written examination.
The two forms
The evaluation form lets you compute an area without summing rectangles: find any antiderivative and subtract its values at the limits. The constant of integration cancels in , so it is omitted in definite integrals. The derivative form says that integrating then differentiating returns the original function.
Evaluating a definite integral
Differentiating an integral function
The derivative form is examined directly when a function is defined by an integral with a variable upper limit. Differentiating simply substitutes the upper limit into the integrand.
Why the two processes are inverse
The theorem shows that the area-accumulation function has the original as its derivative, so accumulating area then measuring its rate of change recovers . Conversely, the evaluation form expresses the net accumulated area as the change in an antiderivative. This duality is the central result of single-variable calculus.