Questions: Find the average value of the function on the given interval. f(x) = sqrt(x+3) ; [6,13]

Solution Steps

Given the function \( f(x) = \sqrt{x + 3} \) and the interval \([6, 13]\).

To find the average value of the function over the interval, we first need to calculate the integral of \( f(x) \) from \( 6 \) to \( 13 \): \[ \int_{6}^{13} \sqrt{x + 3} \, dx \] The result of this integral is: \[ \frac{74}{3} \]

The length of the interval \([6, 13]\) is: \[ 13 - 6 = 7 \]

The average value of the function over the interval is given by: \[ \frac{1}{b - a} \int_{a}^{b} f(x) \, dx \] Substituting the values, we get: \[ \frac{1}{7} \cdot \frac{74}{3} = \frac{74}{21} \]

Evaluating the fraction, we get: \[ \frac{74}{21} \approx 3.5238 \]

Final Answer