The average value of the function f(x) = 5x² on the interval 0 ≤ x ≤ 4 is found by integrating the function over the interval and dividing by the interval length, resulting in an average value of 40.
To find the average value of the function f(x) = 5x² on the interval 0 ≤ x ≤ 4, we integrate the function over the interval and then divide by the length of the interval. The integral of f(x) from 0 to 4 is the area under the curve, which gives us ∫ 5x² dx from 0 to 4. Performing this integration, we get:
- ∫ 5x² dx = ⅓ 5x³ | from x=0 to x=4.
- ⅓ 5(4³) - ⅓ 5(0³) = 160
- The length of the interval is 4 - 0 = 4.
- To find the average value, divide the integral's result by the interval length: Average value = 160 / 4 = 40.
The answer is that the average value of f(x) = 5x² on the interval from 0 to 4 is 40.