Question

Find the average value of f(x)=x⁴ on the interval [2,5]

Answer 1

The average value of the function f(x) = x⁴ on the interval [2,5] is calculated as 206.2 by integrating the function over the interval and dividing by the interval's length.

Step-by-step explanation:

The average value of a function f(x) on a given interval [a, b] can be calculated using the formula:

average value = (1 / (b - a)) × ∫ₓₓ⃒ f(x) dx

For the function f(x) = x4 on the interval [2,5], we can calculate the average value as follows:

1. Calculate the definite integral of f(x) from 2 to 5.
2. Multiply the result by (1 / (5 - 2)).

The definite integral of x4 is (1/5)x5

Evaluating it from 2 to 5 we get:

(1/5)[55 - 25] = (1/5)[3125 - 32] = (1/5)[3093] = 618.6

Now, we multiply by (1 / (5 - 2)):

average value = (1 / 3) × 618.6 = 206.2

Therefore, the average value of f(x) = x4 on the interval [2,5] is 206.2.