Find the average value $h}_{ave$ of the function h on the given interval.

$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),[0,\pi ]$

khi1la2f1qv
2021-11-16
Answered

Find the average value $h}_{ave$ of the function h on the given interval.

$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),[0,\pi ]$

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Mary Darby

Answered 2021-11-17
Author has **11** answers

The given function h and the interval is,

$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),[0,\pi ]$

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

${h}_{ave}=\frac{1}{\pi -0}{\int}_{0}^{\pi}8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right)dx$

$=\frac{8}{\pi}{\int}_{0}^{\pi}{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right)dx$

$=\frac{8}{\pi}{\int}_{1}^{-1}{u}^{4}du$

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

${h}_{ave}=\frac{8}{\pi}{\int}_{-1}^{1}{u}^{4}du$

$=\frac{8}{\pi}\left(\frac{2}{5}\right)$

$=\frac{16}{5\pi}$

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

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

Drood1980

Answered 2021-11-18
Author has **16** answers

Consider the function

$h\left(x\right)=8{\mathrm{cos}}^{4}x\mathrm{sin}x$

The average value of the function in the interval$[0,\pi ]$ is given by

${h}_{ave}=\frac{1}{\pi -0}{\int}_{0}^{\pi}h\left(x\right)dx$

Hence,

${h}_{ave}=\frac{1}{\pi -0}{\int}_{0}^{\pi}8{\mathrm{cos}}^{4}x\mathrm{sin}xdx$

${h}_{ave}=\frac{1}{\pi}{\int}_{-1}^{1}8{t}^{4}dt$

$h}_{ave}=\frac{8}{5\pi}{\left[{t}^{5}\right]}_{-1}^{1$

$h}_{ave}=\frac{16}{5\pi$

Hence, the average value of the functions is$h}_{ave}=\frac{16}{5\pi$

The average value of the function in the interval

Hence,

Hence, the average value of the functions is

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