# Riemann sums

This is one of the more complex examples. It draws Riemann partitions from point a to point b. While computing the partitions, their areas are summed up to show how the Riemann approximation compares to the true area under the given curve.

In this example, some extra markup is used outside of Mafs to provide some inputs into the Mafs visualization. This is common, and recommended! Movable points are not the only way to provide inputs to Mafs.

Partitions:
import * as React from "react"
import { Mafs, useMovablePoint, Plot, Polygon, Text, Coordinates } from "mafs"

import sumBy from "lodash/sumBy"
import range from "lodash/range"

interface Partition {
polygon: [number, number][]
area: number
}

function RiemannSum() {
const maxNumPartitions = 200

// Inputs
const [numPartitions, setNumPartitions] =
React.useState(40)
const lift = useMovablePoint([0, -1], {
constrain: "vertical",
})
const a = useMovablePoint([1, 0], {
constrain: "horizontal",
})
const b = useMovablePoint([11, 0], {
constrain: "horizontal",
})

// The function
const wave = (x: number) =>
Math.sin(3 * x) + x ** 2 / 20 - 2 + lift.y + 2
const integral = (x: number) =>
(1 / 60) * (x ** 3 - 20 * Math.cos(3 * x)) + lift.y * x

// Outputs
const exactArea = integral(b.x) - integral(a.x)
const dx = (b.x - a.x) / numPartitions
const partitions: Partition[] = range(
a.x,
b.x - dx / 2,
dx,
).map((x) => {
const yMid = wave(x + dx / 2)

return {
polygon: [
[x, 0],
[x, yMid],
[x + dx, yMid],
[x + dx, 0],
],
area: dx * yMid,
}
})

const areaApprox = sumBy(partitions, "area")

return (
<>
<Mafs
viewBox={{ x: [-1, 12], y: [-3, 10] }}
>
<Coordinates.Cartesian subdivisions={2} />

<Plot.OfX y={wave} color="#358CF1" />

{partitions.map((partition, index) => (
<Polygon
key={index}
points={partition.polygon}
fillOpacity={numPartitions / maxNumPartitions}
color={
partition.area >= 0
? "hsl(112, 100%, 47%)"
: "hsl(0, 100%, 47%)"
}
/>
))}

<Text attach="e" x={1.2} y={5.5} size={20}>
Midpoint Riemann sum:
</Text>

<Text attach="e" x={1.2} y={4.5} size={30}>
{areaApprox.toFixed(4)}
</Text>

<Text attach="e" x={1.2} y={3.5} size={20}>
True area:
</Text>

<Text attach="e" x={1.2} y={2.5} size={30}>
{exactArea.toFixed(4)}
</Text>

{lift.element}
{a.element}
{b.element}
</Mafs>

{/* These classnames are part of the Mafs docs website—they won't work for you. */}
<div className="p-4 border-gray-700 border-t bg-black text-white">
Partitions:{" "}
<input
type="range"
min={20}
max={200}
value={numPartitions}
onChange={(event) =>
setNumPartitions(+event.target.value)
}
/>
</div>
</>
)
}