Integrals · Definition

Definite integral

calculusintegrals

Before you start you need to know

Central idea of calculus: measuring curved areas by summing infinitely many ever thinner rectangles.

Definition

The definite integral of \(f\) on \([a,b]\) is the limit of the Riemann sums as \(n\to\infty\). It represents the signed net area between the graph of \(f\) and the \(x\)-axis, from \(x=a\) to \(x=b\).

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^{*})\,\Delta x\]

The definite integral \(\int_a^b f(x)\,dx\) is the limit of Riemann sums as the rectangles become infinitely thin. It measures signed net area: what lies above the axis adds, what lies below subtracts.

Every continuous function on a closed interval is integrable. It also inherits useful properties: linearity, additivity over intervals and monotonicity.

Computing it from the definition is heavy; the next step (the fundamental theorem) connects it with antiderivatives.

It is also called the Riemann integral.

Think about it

What does it mean for \(\int_a^b f\) to be negative?

That, in net terms, there is more area below the x-axis than above it.

Exercises 10

  1. What is \(\displaystyle\int_{-1}^{1} x\,dx\)?
  2. The definite integral \(\displaystyle\int_a^b f(x)\,dx\) with \(f(x)\ge 0\) represents...
  3. What is the correct notation for the definite integral of \(f\) from \(a\) to \(b\)?
  4. What is the definite integral built as a limit of Riemann sums called?
  5. In Riemann sums, what does \(\Delta x\) represent?
  6. What is \(\int_0^2 3\,dx\)?
  7. What is \(\int_1^3 2x\,dx\)?
  8. Is it true that if \(\int_{-1}^{1} f(x)\,dx=0\) then \(f(x)=0\) for all \(x\in[-1,1]\)?
  9. Which property lets you split the integral of a sum of functions?
  10. If \(\int_0^3 f(x)\,dx=10\) and \(\int_0^1 f(x)\,dx=4\), what is \(\int_1^3 f(x)\,dx\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Tom M. Apostol, Calculus, Vol. 1, 2.ª edición, John Wiley & Sons, 1967.