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The definite integral — Mathematics, 14–17 years

A way to add up infinitely many tiny contributions, often finding the area under a curve or total accumulated change.

Adding tiny slices

A definite integral adds up quantities spread continuously across an interval. For area, imagine cutting the region under a curve into very thin vertical strips, finding each strip’s area, and adding them. As the strips become thinner, the estimate approaches one exact total.

The problem of changing rates

Ordinary multiplication finds a total when every small part is alike, such as distance = constant speed × time. But a changing speed, uneven land or varying flow has no single value to multiply. Integration was developed to find the total effect of a quantity whose rate changes from place to place or moment to moment.

Area under y = x²

Find the area under y = x² from x = 0 to x = 2. An antiderivative is x³/3, so evaluate it at the ends: 2³/3 − 0³/3 = 8/3. Thus the area is 8/3 square units, about 2.67. The integral notation records this accumulation without drawing every tiny strip.

Area is not always positive

A common mistake is to call every definite integral an ordinary geometric area. The integral counts regions above the horizontal axis positively and regions below it negatively, because it measures signed accumulation. If you need total geometric area, split at the crossings and use positive values for each separate region.

Totals from changing quantities

Integration appears when a total is built from a changing rate: distance from changing speed, mass from changing density, water collected from changing flow, or energy from changing power. It also estimates areas and volumes in science and engineering. The method is useful only when the quantity really varies continuously or can be modelled that way.

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