Riemann Sums & The Definite Integral Cheat Sheet

Key Facts

• For $n$ equal subintervals on $[a,b]$, the width is $\Delta x = \frac{b-a}{n}$.
• A general Riemann sum is $\sum_{i=1}^{n} f(x_i^*)\Delta x$, where $x_i^*$ is a sample point in the $i$th subinterval.
• The left Riemann sum is $L_n = \sum_{i=1}^{n} f(a+(i-1)\Delta x)\Delta x$.
• The right Riemann sum is $R_n = \sum_{i=1}^{n} f(a+i\Delta x)\Delta x$.
• The midpoint Riemann sum is $M_n = \sum_{i=1}^{n} f\left(a+\left(i-\frac{1}{2}\right)\Delta x\right)\Delta x$.
• The definite integral is $\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\Delta x$ when the limit exists.
• If $F'(x)=f(x)$, then the Fundamental Theorem of Calculus gives $\int_a^b f(x)\,dx = F(b)-F(a)$.
• The net area $\int_a^b f(x)\,dx$ counts area above the $x$-axis as positive and area below the $x$-axis as negative.

Vocabulary

Riemann sum

A sum $\sum_{i=1}^{n} f(x_i^*)\Delta x$ that approximates the signed area under a curve using rectangles.

Definite integral

The limit of Riemann sums on an interval, written $\int_a^b f(x)\,dx$, representing net accumulation.

Subinterval

One smaller interval formed when $[a,b]$ is divided into $n$ pieces, each with width $\Delta x$ for equal partitions.

Sample point

The chosen input $x_i^*$ inside a subinterval where the function height $f(x_i^*)$ is measured.

Net area

The signed area found by adding positive regions above the $x$-axis and negative regions below the $x$-axis.

Antiderivative

A function $F(x)$ whose derivative is $f(x)$, so $F'(x)=f(x)$.