Find the average value of the function $ f(x) = \sin^2 x \cos^3 x $ on the interval $ [-\pi, \pi] $.
this problem is from chapters happen section to problem number fifty five of the book Calculus Early Transcendental Sze eighth Edition by James Door Here we like to find the average value of the function sine squared times cosign cube on the interval from negative pile a pie. So in this case, we have a formula for the average valuable function. It's one over B minus, a times the integral from A to B of FX sonar problem. We have average value and I have one over B which is pie minus you too, Pai in a rule A to be so negative pine a pie and then after Becks. So now we have a triggered a metric and rule so on. This General observed that the power of coastline is odd, so let's pull out one factor of co sign so that we could eventually use a new substitution. We have one over to pie negative Pika pi and the integral Science where Coastline Square. Since quarterbacks Excuse me, Cause and square of X, Sam's course Innovex. And now we can use a pathetic and identity to rewrite this in terms of sign co sign squared is one minus sign square. So we have one over to buy in a roll. Negative part of fire signed square one minus sense where times cause I'm at this point we see we can apply a u substitution. Let's take you two be scientifics so that do you is cosign of X t X. Also, because we have a definite integral let's change those limits of integration so the lower limit will become Sign a pie negative by which is zero from the circle, and the upper limit will become a sign of pie from the unit circle. That's also zero. So here are interval becomes one over to pie integral from zero zero you square one minus use where to you And in this case, we don't need to integrate because the answer it'LL be zero because we have an integral from zero two zero. The end points are the same. So in this case, the integral will always evaluate zero. So we have one over to pie time zero, and that's equal to zero. And there's our our answer. There's our average value