Find the average value of the function f(x)=4−x3 on the interval [2,4] . (1 point)
To find the average value of the function f(x) on the interval [a,b], we need to evaluate the definite integral of f(x) over that interval and then divide by the length of the interval.
In this case, f(x) = 4 - x^3 and the interval is [2,4].
The definite integral of f(x) over the interval [2,4] is given by:
∫[2,4] (4 - x^3) dx
To integrate this function, we use the power rule for integration:
∫x^n dx = (1/(n+1)) * x^(n+1) + C
Applying the power rule to each term in the function, we get:
∫[2,4] (4 - x^3) dx = ∫[2,4] 4 dx - ∫[2,4] x^3 dx
= (4x | [2,4]) - ((1/4) * x^4 | [2,4])
= (4 * 4 - 4 * 2) - ((1/4) * (4^4) - (1/4) * (2^4))
= 16 - (64/4 - 16/4)
= 16 - (48 - 4)
= 16 - 44
= -28
So, the definite integral of f(x) over the interval [2,4] is -28.
The length of the interval [2,4] is 4 - 2 = 2.
Therefore, the average value of f(x) on the interval [2,4] is:
(-28) / 2 = -14
So, the average value of the function f(x) = 4 - x^3 on the interval [2,4] is -14.