Integration definition
Integration is a method in calculus used to find accumulation, total change and the area under a curve. It is one of the two main parts of calculus, alongside differentiation. Differentiation studies rates of change, while integration often studies totals built up from changing quantities. Integration can also be described as reverse differentiation because it can undo the effect of differentiating a function.
A simple example is the function 2x. If you know that differentiating x squared gives 2x, then integrating 2x gives x squared plus a constant. The constant is written as C because many different functions can have the same derivative. For example, x squared plus 3 and x squared minus 10 both differentiate to 2x. The constant of integration represents this missing fixed value.
Integration is mainly studied at A-Level and beyond, but it relies on earlier skills from GCSE maths, especially algebra, powers, graphs, area and substitution. Students need to understand functions and be comfortable rearranging expressions. Integration appears in pure maths, mechanics, physics, engineering, economics, biology, statistics and any subject where a total is built from changing rates.
A common integration rule is that the integral of x to the power n is x to the power n plus 1 divided by n plus 1, plus C, as long as n is not -1. For example, the integral of x squared is x cubed divided by 3 plus C. This rule is the reverse of the power rule for differentiation. It helps students integrate many polynomial expressions efficiently.
Definite integration is used to calculate a numerical total or area between two limits. For example, an integral from x = 1 to x = 4 gives a value after substituting the upper limit and subtracting the value from the lower limit. This can represent the exact area under a curve between those two x-values. If the curve is below the x-axis, the integral may give a negative signed area, so interpretation matters.
Integration is important in mechanics. If velocity is a function of time, integrating velocity can give displacement. If acceleration is integrated, it can give velocity. This makes integration useful for motion problems where quantities change continuously. It also appears in problems about distance travelled, work done, growth, decay and accumulated cost over time.
Students sometimes think integration is only about adding one to the power and dividing. That rule is important, but the deeper meaning is accumulation. Integration helps turn many tiny changes into a total result. This is why it is powerful for modelling real situations, such as total rainfall, distance travelled from changing speed, or total profit from a changing rate of sales.
Related ideas include calculus, differentiation, derivative, area under a curve, indefinite integral, definite integral, constant of integration, limits, velocity, displacement and functions. A clear definition of integration helps students understand why the method works and where it can be used. In exams, students should include plus C for indefinite integrals and handle limits carefully for definite integrals.