How do you find the average value of the function for #f(x)=2xsqrt(1+x^2), -3<=x<=3#?
To find the average, we take the integral and divide through the length of the interval.
Now evaluate between -3 and 3 :
F(3) - F(-3) = 0
So 0/6 = 0.
The average value of a function over an interval is equal to the definite integral of that interval divided by the length of the interval. If we wanted to find the average value of
If we plug in our function, we get:
Let's first start by computing the anti-derivative of the function. We can quite quickly see that we have the derivative of
Now we can evaluate the definite integral:
Now we divide by
So, the average value of the function is
In fact, since the function is mirrored upside down on the other side of the x-axis you can say that for any real number