The Area Problem

Main idea

We only know how to find the area of shapes with straight sides, so slice the region under a curve into thin rectangles and add them up. Thinner rectangles hug the curve better, and in the limit the estimate becomes the exact area.

A \approx {\htmlData{sym=2}{\sum_{i=1}^{n}}} {\htmlData{sym=1}{f(x_i^*)}}\,{\htmlData{sym=0}{\Delta x}}, \qquad {\htmlData{sym=0}{\Delta x}} = \frac{b - a}{n}

What the symbols mean

$\Delta x$
Width
Each rectangle's width: the interval split into n equal pieces.
$f(x_i^*)$
Height
The function's value at a sample point in the ith piece: left end, right end or midpoint.
$\sum_{i=1}^{n}$
Sum
Add up all n rectangle areas.

Explanation

The area problem is simply the question: how much space is there between a curve and the x-axis? All we're really doing is approximating that curvy region with rectangles, because a rectangle's area is just width times height. The formula looks dense, but it's only three pieces: $\Delta x$ (the width of each rectangle, the interval chopped into $n$ equal pieces), $f(x_i^*)$ (the height of the $i$ th rectangle, read off the curve) and $\sum$ (add them all up). Width times height, added up. That's a Riemann sum.

This should feel familiar. Back when we defined the derivative, we couldn't find the slope of a curve directly, so we used slopes of straight secant lines and took a limit. Area works the same way: we can't measure a curvy region directly, so we use straight-sided rectangles and take a limit.

Split $[a, b]$ into $n$ equal pieces of width $\Delta x$ . On each piece, build a rectangle whose height is $f$ at a sample point:

Left endpoints, right endpoints, or midpoints.

For an increasing function, left endpoints underestimate and right endpoints overestimate; for a decreasing one it's the reverse. It makes sense if you picture it: on a rising curve, the left end of each piece is its lowest point, so every rectangle sits a bit under the curve. Midpoints are usually the most accurate, because the overshoot on one side of each piece roughly cancels the undershoot on the other.

As $n \to \infty$ , every choice converges to the same number: the exact area. The next lesson names that limit.

See it

These are rectangles under y = x² on [0, 2]. With only a few rectangles the estimate is rough, but as n grows they hug the curve more and more closely, and the total closes in on the exact area, 8/3 ≈ 2.667.

A rooftop solar panel's power over a 12-hour day, P(x) = x(12 − x)/9 kW, peaking at 4 kW at noon. The energy it makes is the area under this curve, and each rectangle is one block of time at a constant power reading. Two rectangles give 24 kWh, four give 30, ten give 31.68. In fact this sum works out to 32(1 − 1/n²), so it closes in on the exact 32 kWh, and by n = 20 it's within 0.08.

Worked examples

Example 1

Estimate the area under $f(x) = x^2$ on $[0, 2]$ with 4 rectangles and right endpoints.

1
Four equal pieces of $[0, 2]$ give $\Delta x = \frac12$ , and the right endpoints are $0.5, 1, 1.5, 2$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Estimate the area under $x^2$ on $[0, 2]$ with 4 rectangles and left endpoints.

Problem 2+5 XP

Now use 4 rectangles and midpoints.

Problem 3+5 XP

Estimate the area under $\frac1x$ on $[1, 3]$ with 2 rectangles and right endpoints.

Problem 4+5 XP

For an increasing function, do left endpoints overestimate or underestimate the area?

Problem 5+5 XP

Evaluate $\displaystyle\sum_{i=1}^{4} i^2$ .

Problem 6+5 XP

A smart meter records a home's power use every $2$ hours: $1.2$ , $0.8$ , $2.5$ and $3.1$ kW at $t = 0, 2, 4, 6$ . Estimate the energy used from $t = 0$ to $t = 8$ hours with a left-endpoint sum.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the exact area under $y = x^2$ on $[0, 2]$ as the limit of right-endpoint sums $R_n$ as $n \to \infty$ .

Where it's used

Your phone estimates the distance you walked by adding speed × time over tiny intervals: a Riemann sum running in real time.

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