Find the average value of the following function on the given interval. Graph the function and indicate the average value. \( f(x)=x^{3} \) on \( [-5,5] \) The average value of the function is \( \mathrm{f}=\square \) (Simplify your answer.) (

To find the average value of a function \( f(x) \) on a given interval \([a, b]\), we use the formula: \[ \text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \] Given: - Function: \( f(x) = x^{3} \) - Interval: \([-5, 5]\) First, we need to find the integral of \( f(x) \) from -5 to 5: \[ \int_{-5}^{5} x^{3} \, dx \] Next, we will calculate the integral and then find the average value of the function on the given interval. Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{-5}^{5} x^{3} dx\) - step1: Evaluate the integral: \(\int x^{3} dx\) - step2: Evaluate the integral: \(\frac{x^{3+1}}{3+1}\) - step3: Add the numbers: \(\frac{x^{4}}{3+1}\) - step4: Add the numbers: \(\frac{x^{4}}{4}\) - step5: Return the limits: \(\left(\frac{x^{4}}{4}\right)\bigg |_{-5}^{5}\) - step6: Calculate the value: \(0\) The integral of \( f(x) = x^{3} \) from -5 to 5 is 0. Now, we can find the average value of the function on the interval [-5, 5] using the formula: \[ \text{Average Value} = \frac{1}{5-(-5)} \times 0 \] \[ \text{Average Value} = \frac{1}{10} \times 0 = 0 \] Therefore, the average value of the function \( f(x) = x^{3} \) on the interval [-5, 5] is 0.