What is the average value of the function $f\left(x\right)=x^2$ on the interval $[0,2]$?

• A. $\frac{2}{3}$
• B. $2$
• C. $\frac{4}{3}$
• D. $1$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function f(x) on the interval [a, b]. It is given by: . Here, a = 0, b = 2, and f(x) = x².
2. Step 2: Substitute the values of a, b, and f(x) into the formula. This gives: . Simplify the denominator to get: .
3. Step 3: Compute the integral $ext\{\int \}\_\{0\}^\{2\}\; x²\; dx$. The antiderivative of x² is . Apply the Fundamental Theorem of Calculus to evaluate the definite integral: .
4. Step 4: Simplify the result of the definite integral. This gives: .
5. Step 5: Multiply the result of the integral by to find the average value: . Simplify the expression to complete the calculation.