Mean Value Theorem For Integrals

How to use

• Drag the blue points $a$ and $b$ along the $x$-axis to choose the interval.
• The blue curve is $f(x)$.
• The blue shaded region represents the area $\displaystyle \int_a^b f(x)\,dx$.
• The red rectangle has base $(b-a)$ and height $\overline f$ (the average value of $f$ on $[a,b]$).
• The blue point labeled $f(\xi)$ is placed so that $f(\xi)=\overline f=\frac{1}{b-a}\int_a^b f(x)\,dx.$ A dashed vertical segment drops from $f(\xi)$ to the $x$-axis to help you read the corresponding $\xi$.
• The gray dashed horizontal lines labeled $m$ and $M$ indicate (an approximation of) the minimum and maximum of $f$ on $[a,b]$. They highlight the inequality $m \le \overline f \le M,$ which guarantees that the horizontal line $y=\overline f$ intersects the graph in the interval.
• Move $a$ and $b$ and watch how the blue shaded area $\int_a^b f(x)\,dx$ changes.
• Notice that the red rectangle adjusts its height so that its area matches the integral area.
• Track the point $f(\xi)$: it always lies on the graph and on the level $y=\overline f$, illustrating the existence of a mean value point.
• Try different interval lengths and positions: the average value $\overline f$ moves up/down, and the displayed $\xi$ shifts accordingly.

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Theorem7.9 page 132 and Figure 7.4

The applet visualizes the Mean Value Theorem for Integrals.

Geometrically, the theorem states that the area of the rectangle with width $(b-a)$ and height $f(\xi)$ (shown in red) is exactly equal to the area under the curve (the integral).

We use a strictly invertible cubic function.

// f(x) = 0.1x³ + 1
const f = (x: number) => 0.1 * Math.pow(x, 3) + 1;

// Antiderivative F(x) = 0.025x⁴ + x
const F = (x: number) => 0.025 * Math.pow(x, 4) + x;

// Analytical Inverse f⁻¹(y) = ∛(10 * (y - 1))
const fInverse = (y: number) => Math.cbrt(10 * (y - 1));

Using the Fundamental Theorem of Calculus, we calculate the integral as $F(b) - F(a)$. We then find the mean value (height) $f(\xi)$ using the theorem:
$f(\xi) = \frac{F(b) - F(a)}{b - a}$

const meanValue = () => {
    const { start, end } = interval();
    if (Math.abs(end - start) < 1e-9) return f(start);
    return (F(end) - F(start)) / (end - start);
};

Since the function is invertible, we find the exact coordinate $\xi$ by applying the inverse function $f^{-1}$ to the calculated mean value. This allows the applet to update the point $f(\xi)$ without relying on numerical methods.

const getXi = () => {
    const target = meanValue();
    return fInverse(target);
};

The $min$ anf $max$ of $f$, inside $[a,b]$, are displayed in gray. Since f(x) is strictly increasing, min is always at start, max is always at end

const extrema = () => {
    const { start, end } = interval();
    return { 
    min: f(start), 
    max: f(end) 
    };
};