Question

Let f(x)=14pix^2 and g(x)=k^2 sin(pix/2k) for k>0. a. Find the average value of f on [1,4]. b. For what value of k will the average value of g on [0,k] be equalto the average value of f on [1,4]?

Answer 1

a. To find the average value, we integrate over the interval/region then divide by the width of the interval. So in this case (I'll leave details to you),you should get the average value to be
98*pi

b. We do basically the same thing here. We want to integrate
k^2 sin(pi*x/2k) dx from 0 to k.
Just keep in mind that k is a constant, and this is pretty easy. Let u=pi*x/2k, so that du=pi/2k * dx. Noting that we're now integrating u from 0 to pi/2,the integral becomes
2/pi * k^3 * sin(u) du,

which evaluates to -2k^3/pi * cos(u), and using our limits, we get 2k^3/pi. Dividing by the width of our interval (k), and equating this to our answer inpart (a), we have that
k=49*pi^2.