How do you find the average value of #f(x)=x/sqrt(x^2+1), 0<=x<=4#?
The average value of
The average value of a continuous function
This now becomes a trig substitution problem. Let
Use the pythagorean identity
This is a known integral.
We know from our initial substitution that
Evaluate this using the 2nd fundamental theorem of calculus.
Hopefully this helps!
Alternative to get
Like HSBC244 said, The average value of a continuous function
Instead of doing a trig substitution like HSBC244, you can do a u substitution instead. Set
Note that the limits of integration changed because we were doing a u substitution. 0 became 1 because:
Similarly plugging in 4 became 17 because:
Now rewrite it so it's easier to integrate by treating the square root in the denominator:
This integral becomes
Take the 2 out and plug in 17 and 1. As you can see, this will give you the same answer as HSBC244's answer but does not require a trig substitution.