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Topic 3: Integral Calculus

4 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we reverse differentiation to recover a function from its rate of change?

Antidifferentiation reverses differentiation to find the family of functions whose derivative is a given function, always including a constant of integration.

How to antidifferentiate power, exponential and reciprocal functions, why the +C constant is essential, and how to find a particular antiderivative from a boundary condition.

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How do we find the exact area between a curve and the x-axis?

The area between a curve and the x-axis equals the definite integral, with regions below the axis requiring a sign adjustment.

How to compute the area between a curve and the x-axis using definite integrals, why regions below the axis need absolute values, and how to handle curves that cross the axis.

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How do we find the area enclosed between two curves?

The area between two curves is the integral of the upper function minus the lower function over the interval where they enclose a region.

How to find the area enclosed between two curves: locate the intersection points, integrate (upper minus lower), and why this method avoids any sign problems.

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How does the definite integral connect antidifferentiation to accumulated change?

The Fundamental Theorem of Calculus evaluates a definite integral as the difference of an antiderivative at the two limits.

How to evaluate definite integrals using the Fundamental Theorem of Calculus, the key properties of definite integrals, and how they represent accumulated change.

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