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khi1la2f1qv

Answered question

2021-11-16

Find the average value

h_{a v e}

of the function h on the given interval.

h (x) = 8 \cos^{4} (x) \sin (x), [0, π]

Mary Darby

Beginner2021-11-17Added 11 answers

The given function h and the interval is,

h (x) = 8 \cos^{4} (x) \sin (x), [0, π]

The average value of the function in the given interval is computed as follows.

h_{a v e} = \frac{1}{π - 0} \int_{0}^{π} 8 \cos^{4} (x) \sin (x) d x

= \frac{8}{π} \int_{0}^{π} \cos^{4} (x) \sin (x) d x

= \frac{8}{π} \int_{1}^{- 1} u^{4} d u

Further evaluate the integral by using the property of the integrals as shown below.

h_{a v e} = \frac{8}{π} \int_{- 1}^{1} u^{4} d u

= \frac{8}{π} (\frac{2}{5})

= \frac{16}{5 π}

Drood1980

Beginner2021-11-18Added 16 answers

Consider the function

h (x) = 8 \cos^{4} x \sin x

The average value of the function in the interval

[0, π]

is given by

h_{a v e} = \frac{1}{π - 0} \int_{0}^{π} h (x) d x

Hence,

h_{a v e} = \frac{1}{π - 0} \int_{0}^{π} 8 \cos^{4} x \sin x d x

h_{a v e} = \frac{1}{π} \int_{- 1}^{1} 8 t^{4} d t

h_{a v e} = \frac{8}{5 π} {[t^{5}]}_{- 1}^{1}

h_{a v e} = \frac{16}{5 π}

Hence, the average value of the functions is

h_{a v e} = \frac{16}{5 π}

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