Definite integrals

Interpret a definite integral as accumulation and connect differentiation with integration.

Meaning

A definite integral is defined as the limit of sums of function values multiplied by small interval widths. For a continuous function, it can be calculated as the difference between an antiderivative's endpoint values. For continuous f, the accumulation A(x) changes by approximately f(x) times a small increase in x, giving A′(x)=f(x). An antiderivative F has the same derivative, so A and F differ by a constant. Subtracting endpoint values cancels it, yielding the integral as F(b)−F(a).

Example

Since the graph is above the horizontal axis, this equals the area of a triangle with base 2 and height 2.Parts where the function is negative contribute negatively to the integral. For total geometric area, split the interval at sign changes or use absolute values.

Practice for this concept

Scope and references

Topics

Accumulation on an interval, properties of definite integrals, the fundamental theorem of calculus

Learning goal

Interpret a definite integral as accumulation and connect differentiation with integration.

Helpful background

Connect accumulation over subdivisions with limits and antiderivatives.