Question

How do you use the integral formula to find the average value of the function $f\left( x \right)=18x$ over the interval between $0$ to $4$?

Answer 1

In this problem we need to calculate the average value of the given function in the given range using the integral formula. We know that the average value of the function $f\left( x \right)$ over the given limits $a$ and $b$ is given by the formula $\dfrac{1}{b-a}\int\limits_{a}^{b}{f\left( x \right)dx}$. In this formula we have the definite integral value of the function. So, we will first calculate the indefinite integral value of the given function by using the integration formulas. After that we will apply the given limits to calculate the definite integral value. Now we will apply this value in the average formula and simplify the obtained equation to get the required solution.

Complete step by step answer:
Given function $f\left( x \right)=18x$.
Given limits are $0$ and $4$.
Integrating the given function to calculate the indefinite integral value, then we will get
$\begin{aligned} & \Rightarrow \int{f\left( x \right)dx}=\int{18xdx} \\\ & \Rightarrow \int{f\left( x \right)dx}=18\int{xdx} \\\ \end{aligned}$
Applying the integration formula $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C$ in the above equation, then we will have
$\begin{aligned} & \Rightarrow \int{f\left( x \right)dx}=18\times \dfrac{{{x}^{2}}}{2}+C \\\ & \Rightarrow \int{f\left( x \right)dx}=9{{x}^{2}}+C \\\ \end{aligned}$
Given limits are $0$ and $4$.
Applying the given limits to the above calculated indefinite integral value to find the definite integral value, then we will get
$\begin{aligned} & \Rightarrow \int\limits_{0}^{4}{f\left( x \right)dx}=9\left[ {{x}^{2}} \right]_{0}^{4} \\\ & \Rightarrow \int\limits_{0}^{4}{f\left( x \right)dx}=9\left( {{4}^{2}}-{{0}^{2}} \right) \\\ & \Rightarrow \int\limits_{0}^{4}{f\left( x \right)dx}=144 \\\ \end{aligned}$
Now the average value of the given function from the integral formula is given by
$\begin{aligned} & \Rightarrow avg=\dfrac{1}{b-a}\int\limits_{a}^{b}{f\left( x \right)dx} \\\ & \Rightarrow avg=\dfrac{1}{4-0}\int\limits_{0}^{4}{f\left( x \right)dx} \\\ \end{aligned}$
Substituting the value $\int\limits_{0}^{4}{f\left( x \right)dx}=144$ in the above equation, then we will get
$\begin{aligned} & \Rightarrow avg=\dfrac{1}{4}\times 144 \\\ & \Rightarrow avg=36 \\\ \end{aligned}$
Hence the average value of the given function in the given limits is $36$.

Note:
In this problem we have the simple function so we can directly use the integration formula to calculate the average value without calculating the values of indefinite and definite integral values. If they have given the complex or composite functions then the above-mentioned method is very useful and gives an error less result.