Integrals · Definition

Riemann sums

calculusintegrals

Before you start you need to know

Riemann sums are the most direct way to approximate areas under a curve before reaching the definite integral.

Definition

A Riemann sum approximates the area under a curve by dividing \([a,b]\) into \(n\) rectangles of width \(\Delta x\) and adding their areas, with the height of each rectangle taken at a point of its subinterval.

\[\begin{gathered}\sum_{i=1}^{n}f(x_i^{*})\,\Delta x\\ \Delta x=\frac{b-a}{n}\end{gathered}\]

A Riemann sum covers the region under \(f\) with rectangles of width \(\Delta x=\frac{b-a}{n}\) and adds their areas. If \(f\) is continuous, increasing \(n\) improves the approximation.

Guiding example: for \(f(x)=x\) on \([0,2]\) with \(n=4\), \(\Delta x=\frac{1}{2}\); using right endpoints the sum is \(\frac{5}{2}\), while left endpoints give \(\frac{3}{2}\).

Typical error: forgetting to multiply by \(\Delta x\) or using the wrong endpoint; the chosen point changes the result, so it must always be specified.

Think about it

What happens to a Riemann sum as n grows?

If the function is integrable, it approaches the area (the definite integral).

Exercises 10

  1. Compute the right-endpoint Riemann sum of \(f(x)=x\) on \([0,2]\) with \(n=4\) equal subintervals.
  2. Compute the left-endpoint Riemann sum of \(f(x)=x\) on \([0,2]\) with \(n=4\) equal subintervals.
  3. Compute the right-endpoint Riemann sum of \(f(x)=x^{2}\) on \([0,3]\) with \(n=3\) equal subintervals.
  4. Compute the right-endpoint Riemann sum of \(f(x)=x^{2} + 1\) on \([0,2]\) with \(n=2\) equal subintervals.
  5. What is the formula for \(\Delta x\) when dividing \([a,b]\) into \(n\) equal subintervals?
  6. Compute \(\Delta x\) for a Riemann sum of \(f\) on \([1,5]\) with \(n=8\) equal subintervals.
  7. Compute the left-endpoint Riemann sum of \(f(x)=4-x\) on \([0,2]\) with \(n=2\) equal subintervals.
  8. Compute the right-endpoint Riemann sum of \(f(x)=x^{2}\) on \([0,2]\) with \(n=4\) equal subintervals.
  9. As \(n\to\infty\), what does a Riemann sum of a continuous function on \([a,b]\) converge to?
  10. Compute the midpoint Riemann sum of \(f(x)=x\) on \([0,2]\) with \(n=2\) equal subintervals.

What comes next

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.