Find the average value of the function f over the interval [-1, 2].
f(x) = 1 - x^2
∫[-1,2] 1-x^2 dx
-----------------------
(2-(-1))
To find the average value of a function over an interval, you need to calculate the definite integral of the function over that interval and divide it by the length of the interval.
In this case, you need to find the definite integral of the function f(x) = 1 - x^2 over the interval [-1, 2].
Step 1: Find the indefinite integral of f(x) = 1 - x^2.
∫(1 - x^2) dx = x - (x^3)/3 + C
Step 2: Evaluate the definite integral over the interval [-1, 2].
∫[-1 to 2] (1 - x^2) dx = [x - (x^3)/3]∣[-1 to 2]
= [2 - (2^3)/3 - (-1 - (-1^3)/3]
= [2 - 8/3 - (-1 + 1/3)]
= 26/3
Step 3: Calculate the length of the interval [-1, 2].
Length = 2 - (-1) = 3
Step 4: Divide the definite integral by the length of the interval to find the average value.
Average value = (1/3) * (26/3) = 26/9
Therefore, the average value of the function f(x) = 1 - x^2 over the interval [-1, 2] is 26/9.