If $f$ is continuous on $[a, b]$ then there exists a number $c$ in $(a, b)$ such that:

\[ \int_a^b f(x) \, dx = f(c)(b – a) \]

Understanding the Mean Value Theorem for Integrals

Hello students! Today, we will learn about the Mean Value Theorem for Integrals. This theorem helps us find the average value of a function over an interval.

What Does the Theorem Say?

If a function $f(x)$ is continuous on the interval $[a, b]$, then there is at least one number $c$ between $a$ and $b$ such that:

\[\begin{align*} \int_a^b f(x) \, dx = f(c) \times (b – a) \end{align*}\]

This means the total area under the curve from $a$ to $b$ equals the area of a rectangle with width $(b – a)$ and height $f(c)$.

Why Is This Important?

It tells us that even if the function changes values, there is some point where the function’s value represents the average over the interval.

Simple Example

Let’s consider $f(x) = x$ on the interval $[2, 5]$.

Step 1: Compute the integral:

\[\begin{align*} \int_2^5 x \, dx = \left[ \dfrac{x^2}{2} \right]_2^5 = \dfrac{25}{2} – \dfrac{4}{2} = \dfrac{21}{2} \end{align*}\]

Step 2: Calculate $(b – a)$:

\[\begin{align*} b – a = 5 – 2 = 3 \end{align*}\]

Step 3: Set up the equation:

\[\begin{align*} \int_2^5 f(x) \, dx = f(c) \times (b – a) \implies \dfrac{21}{2} = f(c) \times 3 \end{align*}\]

Step 4: Solve for $f(c)$:

\[\begin{align*} f(c) = \dfrac{21}{2} \div 3 = \dfrac{7}{2} \end{align*}\]

Step 5: Find $c$ such that $f(c) = \dfrac{7}{2}$:

\[\begin{align*} f(c) = c = \dfrac{7}{2} \end{align*}\]

Since $\dfrac{7}{2} = 3.5$, and $3.5$ is between $2$ and $5$, this satisfies the theorem.

Key Points to Remember

– The function must be continuous on $[a, b]$.
– There exists a $c$ in $(a, b)$ where the theorem holds.
– The theorem connects the integral of a function to its average value.

Conclusion

The Mean Value Theorem for Integrals helps us find where a function’s value equals its average over an interval.

\[ \begin{array}{|c|} \hline \text{\Large Mean Value Theorem for Integrals:} \\ \text{If } f \text{ is continuous on } [a, b], \text{ then there exists a number } c \text{ in } (a, b) \text{ such that:} \\  \int_a^b f(x) \, dx = f(c)(b – a) \\ \hline \end{array} \]