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Riemann Sums
and Notation

Master the bridge from discrete sums to continuous integrals: summation notation, left/right/midpoint sums, and the definite integral.

AP Calculus AB

Why Riemann Sums Matter

How do you find the exact area under a curve when there is no simple geometric formula? The answer: approximate the region with rectangles, then take the limit as the number of rectangles approaches infinity.

The Big Idea: Rectangles Under a Curve

x y a b n = 6 rectangles f(x)

Key Vocabulary

\(\Delta x\)
Width of each sub-interval: \(\Delta x = \frac{b-a}{n}\)
\(n\)
Number of rectangles (sub-intervals) in the partition
\(x_i^*\)
Sample point in the \(i\)-th sub-interval (left, right, or midpoint)
\([a,b]\)
The closed interval over which we approximate or integrate

Summation Notation: \(\sum\)

Sigma notation provides a compact way to write long sums. The general Riemann sum is:

\(i\) is the index of summation. \(1\) is the lower bound and \(n\) is the upper bound. Each term in the sum represents the area of one rectangle.

Three Types of Sample Points

Left Riemann Sum
\(x_i^* = a + (i-1)\Delta x\)
Height at the left endpoint of each sub-interval
Right Riemann Sum
\(x_i^* = a + i\,\Delta x\)
Height at the right endpoint of each sub-interval
Midpoint Sum
\(x_i^* = a + \left(i - \tfrac{1}{2}\right)\Delta x\)
Height at the center of each sub-interval

Left vs Right Riemann Sums

Left Sum Right Sum Overestimate (decreasing f) Underestimate (decreasing f)

The Formal Definition

From Sums to Integrals

Riemann Sum finite n Take Limit \(n \to \infty\) Definite Integral \(\int_a^b f(x)\,dx\)

The definite integral \(\displaystyle\int_a^b f(x)\,dx\) is not a new operation: it is simply the exact value that Riemann sums approach as the number of rectangles grows without bound.

Riemann Sum vs Definite Integral

Riemann Sum Definite Integral
Nature Approximation Exact value
Notation \(\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x\) \(\displaystyle\int_a^b f(x)\,dx\)
Number of rectangles Finite \(n\) \(n \to \infty\) (limit)
Result A number (depends on \(n\)) A single number
Usage Numerical estimation Theoretical / exact

How to Compute a Riemann Sum

1
Find \(\Delta x\): Compute \(\Delta x = \frac{b - a}{n}\).
2
List sample points: Determine \(x_i^*\) for each sub-interval using left, right, or midpoint rule.
3
Evaluate \(f(x_i^*)\): Plug each sample point into the function.
4
Multiply and sum: Compute \(\displaystyle\sum_{i=1}^{n} f(x_i^*)\,\Delta x\).

Example 1: Left Sum of \(f(x) = x^2\)

Approximate \(\displaystyle\int_0^2 x^2\,dx\) using a left Riemann sum with \(n = 4\).

0 0.5 1 1.5 2 0 0.25 1 2.25 \(f(x) = x^2\)

Example 2: Right Sum of \(f(x) = x^2\)

Same integral \(\displaystyle\int_0^2 x^2\,dx\), now with a right Riemann sum, \(n=4\).

Example 3: Midpoint Sum from a Table

A function \(g\) is continuous on \([1, 5]\). Use the table below and a midpoint Riemann sum with 2 equal sub-intervals.

x 1 2 3 4 5
\(g(x)\) 3 7 5 9 2

Definite Integral Notation Decoded

∫ a b \(f(x)\,dx\) lower limit upper limit integrand variable of integration integral sign

Common Mistakes to Avoid

✕ Wrong \(\Delta x\)
Using \(\frac{b}{n}\) instead of \(\frac{b-a}{n}\). Always subtract the lower limit.
✕ Off-by-One Index
Left sums use \(i = 0\) to \(n-1\) (or \(i=1\) with \(x_{i-1}\)). Right sums use \(i = 1\) to \(n\).
✕ Forgetting to Multiply by \(\Delta x\)
Each term is \(f(x_i^*) \cdot \Delta x\), not just \(f(x_i^*)\). The \(\Delta x\) is the rectangle width.

Summary: Notation at a Glance

Concept Notation Meaning
Sub-interval width \(\Delta x = \frac{b-a}{n}\) Width of each rectangle
Left sum \(\displaystyle\sum_{i=0}^{n-1} f(x_i)\Delta x\) Heights at left endpoints
Right sum \(\displaystyle\sum_{i=1}^{n} f(x_i)\Delta x\) Heights at right endpoints
Midpoint sum \(\displaystyle\sum_{i=1}^{n} f(\bar{x}_i)\Delta x\) Heights at midpoints
Definite integral \(\displaystyle\int_a^b f(x)\,dx\) Exact area (limit of sums)

Question 1 of 5

If \(f\) is defined on \([2, 8]\) and we use \(n = 3\) sub-intervals, what is \(\Delta x\)?

Question 2 of 5

Which expression represents a right Riemann sum for \(\displaystyle\int_1^5 f(x)\,dx\) with \(n\) sub-intervals?

Question 3 of 5

Using a left Riemann sum with \(n = 4\) on \([0, 2]\) for \(f(x) = 3x\), what is the approximation?

Question 4 of 5

The expression \(\displaystyle\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x\) is equivalent to which of the following?

Question 5 of 5

For an increasing function on \([a, b]\), which statement is always true?