This lesson mixes the chapter. The tools are the criterion, the two families of integrable functions, the four properties, and the two halves of the Fundamental Theorem — and deciding which applies is part of the work.

A surveyor with four instruments does not pick one at random. Which to use is decided by what is being measured, and these questions practise that decision rather than any one measurement.

The integral is defined by squeezing with upper and lower sums. A bounded function is integrable exactly when some partition makes the gap small. Continuous and monotone functions qualify. The integral adds, scales, splits and respects order, and it is tied to the derivative by the two halves of one theorem.

• The criterion, when integrability is in question.
• The two families, when the function is recognisably continuous or monotone.
• Additivity and the bound $\lvert\int f\rvert\le M(b-a)$, when an integral has to be estimated rather than found.
• The first half of the Fundamental Theorem, when an integral is being differentiated; the second, when one is being evaluated.

Great work! You now know how to:

• Decide integrability by the criterion or by recognising a family
• Evaluate an integral by an antiderivative and estimate one by a bound
• Say which half of the Fundamental Theorem a question needs