Let $f\left(x\right)=x^2+1$. What is the average value of $f$ on the closed interval $[0,2]$?

• A. $3$
• B. $4$
• C. $2.5$
• D. $2$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function f(x) on a closed interval [a, b]. It is given by: . Here, a = 0 and b = 2.
2. Step 2: Substitute the given function f(x) = x^2 + 1 into the formula. The integral becomes: .
3. Step 3: Break the integral into two parts for easier computation: .
4. Step 4: Compute each integral separately. For $ext\{\int \}\; x^2\; dx$, use the power rule: . For $ext\{\int \}\; 1\; dx$, the result is simply x evaluated at the bounds.
5. Step 5: After evaluating the integrals, combine the results and multiply by to find the average value of the function on the interval [0, 2].