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Math Calculators ▶ Average Value of a Function Calculator

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**Table of Content**

Make use of this free average value of a function calculator to find the average of a function over a specified interval. Moreover, you could also get detailed calculations involved in the procedure with this average value over interval calculator.

So without wasting time, let’s get ahead and discuss the method under discussion to understand it better.

Stay focused!

In the context of calculus, the average value of a function over an interval is given as follows:

$$ {f_{avg}} = \frac{1}{{b – a}}\int_{{\,a}}^{{\,b}}{{f\left( x \right)\,dx}} $$

where;

f(x) is a continuous function and \(\left[ {a,b} \right]\) is the interval in which its continuity remains.

If you want to calculate average function’s value manually, you may find it quite difficult. But do not worry as we will be resolving a couple of examples to clarify your concept. For more instance, you can also let this average value of a function calculator to determine average value integral.

**Example # 01:**

How to find the average value of the function over the interval [2, 4] given as under:

$$ x^{2} + 3 x – 6 $$

**Solution:**

By using average value formula calculus:

$$ \bar{f}= \frac{1}{b-a} \int_a^b f \left( x \right) dx $$

$$ L = \frac{1}{\left( 4 \right)-\left( 2 \right)}\int_{2}^{4} x^{2} + 3 x – 6 dx $$

$$ = \int_{2}^{4} \frac{x^{2}}{2} + \frac{3 x}{2} – 3dx $$

As it becomes a definite integral calculator, so you have to simplify it to get the final answer which is:

$$ \text{Average value of the given function} = \frac{37}{3} $$

For steps, tap the integral calculator.

Apart from this, you can also subject to the free mean value theorem calculator to determine the mean value of a function with detailed steps.

**Example # 02:**

How to find the average value of a function as under:

$$ 3x^3 -4x $$

$$ interval = [0, 1]

**Solution:**

Finding the average value of a function given:

$$ \bar{f}= \frac{1}{b-a} \int_a^b f \left( x \right) dx $$

$$ L = \frac{1}{\left( 2 \right)-\left( 0 \right)}\int_{0}^{2} 3 x^{3} – 4 x dx $$

$$ = \int_{0}^{2} \frac{3 x^{3}}{2} – 2 xdx $$

$$ \text{Average value of the given function} = 2 $$

For steps, tap the integral calculator.

Even this free average value of a function calculator also lets you generate accurate results without compromising on computations’ accuracy.

For operating this free calculator, you are required to provide the following input values:

**Input:**

- Write down the function in the designated field
- After that, fetch in the upper and lower limits
- At last, tap the calculate button

**Output:**

The free average value of function calculator does the following calculations:

- Find the average of the given function using the average of a function formula
- Also, displays the detailed calculations involved

The average value of the function is used to analyse graphical interpretation of the function and check its behaviour. Moreover, you can also employ another exponential growth calculator to see any function’s graph along with all parameters defined.

The average value of a function is basically a simple number value that allows us to carry on complex calculus calculations without any hurdle. Otherwise, you have to deal with complicated combinations of variables that may seem very tricky. That is why pupils and professional mathematicians make use of this free average value of a function calculator to make swift calculations and get instant output without compromising the accuracy.

From the source of Wikipedia: Mean of a function, Mean value theorem, Cauchy’s mean value theorem, Implications

From the source of Khan Academy: Average value of function

From the source of Lumen Learning: Average Value of a Function, Definite Integral, Fundamental Theorem of Calculus