Question

Find the average of the function over the given interval and all values of x in the interval for which the function equals its average value. (Round your answer to three decimal places.)
f(x) = 8 cos(x),
(0, π/6)

Answer 1

integrate and then divide by the length of the path

Answer 2

i.e. find
\[\frac{6}{\pi}\int_0^{\frac{\pi}{6}} 8 \cos(x)dx\]

Answer 3

\[f_{average} = \frac{1}{measure(D)}\int_D f\]

Answer 4

So yeah, what satellite said.

Answer 5

easy enough integral, since the antiderivative is just
\[8\sin(x)\] get
\[\frac{6}{\pi} \sin(\frac{\pi}{6})\]
\[\frac{6}{\pi}\times \frac{1}{2}=\frac{3}{\pi}\]

Answer 6

bet this is class presupposes no measure theory though.

Answer 7

oh damn forgot the 8 !

Answer 8

answer is
\[\frac{24}{\pi}\]

Answer 9

now you get to try to solve
\[8\sin(x)=\frac{24}{\pi}\] have fun
start with
\[\sin(x)=\frac{3}{\pi}\] and then maybe graph?

Answer 10

i get some arcsin stuff ugh how do i solve that

Answer 11

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