what is the average value of √x on the interval 5, 12? choose 1 answer: a 2/21 · (√12³ - √5³) b 2/21 ·…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a = 5$, $b = 12$, and $f(x)=\sqrt{x}=x^{\frac{1}{2}}$. So, $\bar{y}=\frac{1}{12 - 5}\int_{5}^{12}x^{\frac{1}{2}}dx=\frac{1}{7}\int_{5}^{12}x^{\frac{1}{2}}dx$.

Step2: Integrate the function

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $n=\frac{1}{2}$, we have $\int x^{\frac{1}{2}}dx=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\frac{2}{3}x^{\frac{3}{2}}+C$.

Step3: Evaluate the definite integral

$\frac{1}{7}\int_{5}^{12}x^{\frac{1}{2}}dx=\frac{1}{7}\left[\frac{2}{3}x^{\frac{3}{2}}\right]{5}^{12}=\frac{2}{21}\left[x^{\frac{3}{2}}\right]{5}^{12}=\frac{2}{21}(\sqrt{12^{3}}-\sqrt{5^{3}})$.

Answer:

A. $\frac{2}{21}\cdot(\sqrt{12^{3}}-\sqrt{5^{3}})$