Find the average value of the function f(x)=4−x3 on the interval [2,4] . (1 point) Responses −52 − 52 −26 − 26 −13 − 13 22 22 54
To find the average value of a function on an interval, we need to calculate the definite integral of the function over that interval and divide it by the length of the interval.
The definite integral of f(x) = 4 - x^3 on the interval [2,4] is:
∫[2,4] (4 - x^3) dx
To evaluate this integral, we can use the power rule of integration. Integrating term by term, we get:
∫[2,4] 4 dx - ∫[2,4] x^3 dx
The first integral is easy to evaluate, it gives us:
4x evaluated from 2 to 4 = 4(4) - 4(2) = 16 - 8 = 8
For the second integral, we use the power rule of integration:
∫ x^n dx = (1/(n+1)) * x^(n+1)
Applying this rule to the integral of x^3, we get:
(1/4) * x^4 evaluated from 2 to 4 = (1/4) * (4^4) - (1/4) * (2^4) = 4 - 1 = 3
Therefore, the definite integral of f(x) = 4 - x^3 on the interval [2,4] is 8 - 3 = 5.
The length of the interval [2,4] is 4 - 2 = 2.
Finally, to find the average value of the function, we divide the definite integral by the length of the interval:
Average value = 5/2 = 2.5.
Therefore, the average value of the function is 2.5.