Two Symbols, One Object

You'll learnto read every part of a sigma expression and of a definite integral, to build a Riemann sum for any n out of the left endpoint and the index, and to match the two notations piece by piece — because the marks in this topic are lost to a misread symbol, not to arithmetic.

A Riemann sum is a list of products added up. A definite integral is what that list becomes when the rectangles get infinitely thin. Both get written in compressed notation, and the compression is where the marks go: an index bound read as a value, a sample point built without the left endpoint, a differential treated as punctuation. Every one of those is a symbol misread rather than a number miscalculated, which is why this topic is worth slowing down for.

8 STEPS · 6 QUICK CHECKS · EVERY PART OF THE SIGMA HAS A COUNTERPART IN THE INTEGRAL · v1

Σ f(x) Δx over n rectangles ⇒ ∫ from a to b of f(x) dx

Before you start

What you're looking at

One plot panel. The blue curve is the integrand, the pink shading is the exact region under it between x = 1 and x = 3, and the yellow rectangles are a right Riemann sum approximating that region. The card column on the right carries the notation.

The question

How does a list of products get written for every n at once, and what does each piece of that expression turn into when the list becomes an integral?

Watch for

Every sum on screen is computed from the same function the curve is drawn from, and the exact value comes from a separate quadrature. The gap between them is printed rather than described.

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