Riemann sums

Definition

Riemann sums approximate the integral of a function using finite sums. They involve partitioning the interval and summing the product of function values and subinterval widths.

5 Must Know Facts For Your Next Test

1. A Riemann sum can be left, right, or midpoint depending on where the sample point is taken within each subinterval.
2. The accuracy of a Riemann sum increases as the number of subintervals increases.
3. The limit of Riemann sums as the number of subintervals approaches infinity gives the exact value of the definite integral.
4. Riemann sums are a fundamental concept in understanding numerical integration techniques like the trapezoidal rule and Simpson's rule.
5. The width of each subinterval in a uniform partition is given by $\Delta x = \frac{b - a}{n}$, where $[a, b]$ is the interval and $n$ is the number of subintervals.

Review Questions

• What are the three types of Riemann sums based on sample points?
• How does increasing the number of subintervals affect the accuracy of a Riemann sum?
• What mathematical concept do you approach by taking the limit of Riemann sums as the number of subintervals goes to infinity?

Related terms

Definite Integral: Represents the signed area under a curve from point $a$ to point $b$ and can be approximated using Riemann sums.

Trapezoidal Rule: A technique for numerical integration that approximates the region under a curve as a series of trapezoids.

Simpson's Rule: A method for numerical integration that approximates functions using parabolic segments for more accurate results.