Appilcations of Integration

Question

Find the average value of f on the given interval.

f(x)=x^2 for 0≤x≤6

Answer 1

average value of a function, f(x), over an interval [a,b], is given by (1/(b - a))[integral from a to b of f(x)dx], which in this case, a=0 and b=6, becomes [1/(6 - 0)]S(0,6)x^2 dx = (1/6)[x^3/3](0,6) = (1/18)[6^3 - 0^3] = (1/18)(6)(36) = 36/3 = 12

Answer 2

The average value of an integrable function f(x) on an interval [a,b] is                    (1/(b-a))[integral from a to b of f(x) dx.

 

So, the average value of x² on [0, 6] is:

 

    (1/(6-0))[integral from 0 to 6 of x² dx] = (1/6)(6³/3) = 12