Verified answer

The average value of f(x) on the interval [a,b] is equal to the integral of f(x) on the interval divided by the length of the interval.

So, the average value of f(x) = definite integral of (1+x)^(1/2) dx divided by 4.

The definite integral of (1+x)^(1/2)dx = (2/3)(1+x)^(3/2) evaluated between -1 and 3 = 16/3. Now divide this by 4 to get the average value of 4/3.

Average value of some function y = f(x) on the interval [a,b] is defined as:

1/(b - a) * ∫f(x) dx from a to b.

In this case:

Avg. value = 1/4 * ∫√(1 + x) dx from -1 to 3

= 1/6 * ((1 + x)^(3/2) eval. from -1 to 3)

= 1/6 * (8 ) = 4/3

The explanation became not mandatory. The mathematical notation is plenty clearer than the spelled-out version. besides, meaning you combine g(x) from 0 to 2, and divide via the size of the era = 2 - 0 = 2. easiest thank you to do the mixture is to make the substitution u = a million + x^3, du = 3x^2 dx so x^2 dx = (a million/3) du and necessary x^2 sqrt(a million + x^3) dx = necessary (a million/3) u^(a million/2) du