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The Anatomy of a Slice

To estimate the area under a curve $f(x)$ between point $a$ and point $b$ , we chop the region into $n$ vertical rectangles.

First, we calculate the width of each rectangle, which we call $\Delta x$ (delta x):

\Delta x = \frac{b - a}{n}

Next, we need the height of the rectangle. But the curve is slanted at the top! Where do we measure the height from? We have three primary choices:

1. Left Sum

Measure the height from the left corner of the rectangle. Good for decreasing functions.

2. Right Sum

Measure from the right corner. Good for increasing functions.

3. Midpoint Sum

Measure from the exact middle. Usually the most accurate approximation.

Approximation Engine

Parameters

Rectangles (n)5

Width (Δx)1.60

Calculations

Est. Area33.088

Exact Area33.600

Error Margin0.512

Sigma Notation

To get the total area, we sum up the area of every rectangle (Height × Width). Mathematicians use the Greek letter Sigma ( $\Sigma$ ) to represent a large sum.

\text{Area} \approx \sum_{i=1}^{n} f(x_i) \Delta x

Let's break the formula down:

$f(x_i)$ The Height of rectangle $i$ .
$\Delta x$ The Width of the rectangle.
$\Sigma$ "Add them all up, starting from rectangle 1 to rectangle $n$ ."

As you saw in the lab, when $n = 5$ , the estimate is rough. But when $n = 50$ , it gets incredibly close. What happens when $n = \infty$ ? That is the gateway to the Definite Integral.