How do you find the average value of #f(x)=x/sqrt(x^2+1), 0<=x<=4#?
2 Answers
The average value of
Explanation:
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
Explanation:
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.