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Riemann Sums
& Integral Notation

Master the bridge from discrete approximations to exact areas using summation notation and the definite integral.

AP Calculus BC

What Is a Riemann Sum?

A Riemann sum approximates the area under a curve by dividing the region into thin rectangles, computing each rectangle's area, and adding them up. As we use more and more rectangles, the approximation approaches the exact area.

x y a b 6 rectangles approximating area under f(x)

Anatomy of a Riemann Sum

Every Riemann sum has three ingredients: the interval \([a, b]\), the number of subintervals \(n\), and a sample point rule for choosing where to evaluate \(f\).

Subinterval Width
\(\Delta x = \dfrac{b - a}{n}\)
Sample Point
\(x_i^*\) in the \(i\)-th subinterval
Rectangle Area
\(f(x_i^*) \cdot \Delta x\)

Three Sample Point Rules

The choice of sample point \(x_i^*\) within each subinterval determines the type of Riemann sum. Each choice gives a different approximation for the same function.

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

Summation Notation: \(\sum\)

Sigma notation is compact shorthand for adding many terms that follow a pattern. The variable \(i\) is the index, running from the lower bound to the upper bound.

∑
\(n\)
 upper bound
\(i=1\)
 index starts here
\(f(x_i^*) \cdot \Delta x\)
 expression evaluated for each i "Add up all rectangle areas"

Key Summation Properties

1
Constant multiple: \(\displaystyle\sum_{i=1}^{n} c\,a_i = c\sum_{i=1}^{n} a_i\)
2
Sum/difference: \(\displaystyle\sum_{i=1}^{n}(a_i \pm b_i) = \sum_{i=1}^{n}a_i \pm \sum_{i=1}^{n}b_i\)
3
Constant sum: \(\displaystyle\sum_{i=1}^{n} c = n \cdot c\)
4
Useful formula: \(\displaystyle\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\)

The Definite Integral as a Limit

The definite integral is defined as the limit of Riemann sums as the number of rectangles approaches infinity:

\(\int\)
Elongated "S" for sum
\(a\) and \(b\)
Limits of integration (lower, upper)
\(f(x)\)
The integrand: the function being integrated
\(dx\)
Indicates the variable; corresponds to \(\Delta x\)

Riemann Sum vs. Definite Integral

Riemann Sum Definite Integral
Symbol \(\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x\) \(\displaystyle\int_a^b f(x)\,dx\)
Rectangles Finite number \(n\) Infinite (\(n \to \infty\))
Result Approximation Exact value
Width \(\Delta x = \frac{b-a}{n}\) \(dx\) (infinitesimal)
When to use Numerical estimates, tables Exact area, FTC evaluation

How to Compute a Riemann Sum

1
Identify the interval \([a,b]\) and the number of subintervals \(n\).
2
Compute \(\Delta x = \dfrac{b - a}{n}\).
3
Determine sample points \(x_i^*\) (left, right, or midpoint).
4
Evaluate \(f(x_i^*)\) for each \(i = 1, 2, \ldots, n\).
5
Sum it up: \(\displaystyle S_n = \sum_{i=1}^{n} f(x_i^*)\,\Delta x\).

Example 1: Right Riemann Sum

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

x y 0 0.5 1 1.5 2 0.25 1 2.25 4

Example 2: Summation to Integral

Write the following Riemann sum as a definite integral:

1
Identify \(\Delta x\): \(\frac{2}{n}\), so \(b - a = 2\).
2
Identify \(x_i^*\): \(3 + \frac{2i}{n} = a + i\,\Delta x\), so \(a = 3\) and \(b = 5\).
3
Identify \(f(x)\): The expression inside is \((x_i^*)^2\), so \(f(x) = x^2\).

Example 3: Evaluating with Formulas

Evaluate \(\displaystyle\int_0^3 (2x + 1)\,dx\) using the limit definition with right endpoints.

1
Setup: \(\Delta x = \frac{3}{n}\), \(x_i = 0 + i \cdot \frac{3}{n} = \frac{3i}{n}\).
2
Substitute: \(f(x_i) = 2\left(\frac{3i}{n}\right) + 1 = \frac{6i}{n} + 1\).
3
Sum: \(\displaystyle\sum_{i=1}^{n}\left(\frac{6i}{n} + 1\right)\frac{3}{n} = \frac{3}{n}\sum_{i=1}^{n}\frac{6i}{n} + \frac{3}{n}\sum_{i=1}^{n}1\).
4
Simplify: \(= \frac{18}{n^2}\cdot\frac{n(n+1)}{2} + 3 = 9\cdot\frac{n+1}{n} + 3\).
5
Take the limit: \(\displaystyle\lim_{n\to\infty}\left(9\cdot\frac{n+1}{n} + 3\right) = 9 + 3 =\) 12.

Over and Under Estimates

Whether a Riemann sum overestimates or underestimates depends on the function's behavior and the sample point rule.

Left Sum Right Sum
\(f\) increasing on \([a,b]\) Underestimate Overestimate
\(f\) decreasing on \([a,b]\) Overestimate Underestimate

Integral Notation Conventions

1
Order of limits: \(\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\). Swapping limits negates the integral.
2
Zero-width interval: \(\displaystyle\int_a^a f(x)\,dx = 0\). No width means no area.
3
Additivity: \(\displaystyle\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx\).
4
Linearity: \(\displaystyle\int_a^b [c\,f(x) + g(x)]\,dx = c\int_a^b f(x)\,dx + \int_a^b g(x)\,dx\).

Common Mistakes

✕ Wrong \(\Delta x\)
Using \(\frac{b}{n}\) instead of \(\frac{b-a}{n}\). Always subtract the lower limit from the upper.
✕ Index Off-by-One
Confusing left (\(i-1\)) and right (\(i\)) sample points, or starting the index at 0 when the formula starts at 1.
✕ Forgetting the Limit
Writing \(\sum\) without \(\lim_{n\to\infty}\) when asked for the definite integral. A finite sum is only an approximation.

Summary of Key Formulas

Concept Formula / Notation
Subinterval width \(\Delta x = \dfrac{b - a}{n}\)
Right endpoint \(x_i = a + i\,\Delta x\)
Left endpoint \(x_i = a + (i-1)\,\Delta x\)
Riemann sum \(\displaystyle\sum_{i=1}^{n}f(x_i^*)\,\Delta x\)
Definite integral \(\displaystyle\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\,\Delta x\)
Sum of first \(n\) integers \(\displaystyle\sum_{i=1}^{n}i = \frac{n(n+1)}{2}\)
Sum of first \(n\) squares \(\displaystyle\sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}\)

Question 1 of 5

If \([a,b] = [1, 5]\) and \(n = 8\), what is \(\Delta x\)?

Question 2 of 5

Which expression represents the right Riemann sum for \(\displaystyle\int_0^3 f(x)\,dx\) with \(n\) subintervals?

Question 3 of 5

The limit \(\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\sqrt{1+\frac{4i}{n}}\cdot\frac{4}{n}\) equals which definite integral?

Question 4 of 5

For a function \(f\) that is increasing on \([2, 6]\), which statement is true?

Question 5 of 5

Evaluate: \(\displaystyle\sum_{i=1}^{4}\bigl(2i - 1\bigr)\)