Answer
$\frac{5}{8}$
Work Step by Step
Step 1. Given the function $f(x)=1-(cos\frac{\pi t}{4})^4$ and the interval $[0,4]$, we can divide the interval into four parts of equal width of $1$ as shown in the figure. And we can find the middle points of each part as $t=\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}$
Step 2. We can find the height of each box (figure not to scale) as
$f(\frac{1}{2})=1-(cos\frac{\pi}{8})^4=1-(\frac{1+cos\frac{\pi}{4}}{2})^2=\frac{5-2\sqrt 2}{8}$,
$f(\frac{3}{2})=1-(cos\frac{3\pi}{8})^4=1-(\frac{1+cos\frac{3\pi}{4}}{2})^2=\frac{5+2\sqrt 2}{8}$,
$f(\frac{5}{2})=1-(cos\frac{5\pi}{8})^4=1-(\frac{1+cos\frac{5\pi}{4}}{2})^2=\frac{5+2\sqrt 2}{8}$,
$f(\frac{7}{2})=1-(cos\frac{7\pi}{8})^4=1-(\frac{1+cos\frac{7\pi}{4}}{2})^2=\frac{5-2\sqrt 2}{8}$
Step 3. We can approximate the area under the function over this interval as the sum of the four rectangles $A=1\cdot (f(\frac{1}{2})+f(\frac{3}{2})+f(\frac{5}{2})+f(\frac{7}{2}))=\frac{5}{2}$
Step 4. The average value of the function over the interval is the area divided by the interval range; thus we have $\bar f_{[0,4]}=\frac{A}{4-0}=\frac{5}{8}$
