Calculus I · Unit 3A · lesson

Partitions and Sigma Notation

Concept

Learning objectives

Divide an interval into subintervals, calculate $\Delta x$ , and read or write finite sums in sigma notation.

Partitions and Sigma Notation

Explanation

Turning an interval into organized data

A partition is simply a list of points that cuts an interval into smaller subintervals. The widths need not be equal, although equal widths make many hand calculations easier. Once the pieces are named, sigma notation lets us describe repeated arithmetic without writing a page-long sum. The notation is compressed, but the underlying action is ordinary: evaluate, multiply by a width, and add.

Understanding the indices matters more than memorizing the symbol. The lower and upper limits of a sum tell us which terms are included, while the formula beside the sigma tells us how the $i$ -th term is formed. In Riemann sums, the index connects each sample value to its corresponding subinterval width. A student who can expand a sigma expression into several concrete terms is far less likely to lose an endpoint or use the wrong number of rectangles.

A partition divides $[a,b]$ into smaller intervals. For $n$ equal pieces,

\Delta x=\frac{b-a}n, \qquad x_i=a+i\Delta x.

The symbol $\sum$ compresses repeated addition:

\sum_{i=1}^n a_i=a_1+a_2+\cdots+a_n.

Guided walkthrough

Build a uniform partition

Partition $[1,5]$ into $8$ equal subintervals.

Answer reveal

Worked solution

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What is $\Delta x$ when $[2,8]$ is divided into $10$ equal subintervals?

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Use $(b-a)/n$ .

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Exercise

Expand $\sum_{i=1}^4(2i-1)$ .

Exercise

Write $3^2+4^2+5^2+6^2$ in sigma notation.

Exercise

Find the left, right, and midpoint sample points for four equal subintervals of $[0,2]$ .

After the explanation

Use the section idea

Reading lens

Treat a definite integral as the limit of structured approximations, with the sample rule and sign visible in every rectangle.

Mental model

Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.

Decision

Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.

Common trap

Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.

Check yourself

Can you construct the sum from a table or formula and predict whether it is high or low before calculating?

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