 # Find the average value h_{ave} of the function h on the given interval. NS khi1la2f1qv 2021-11-16 Answered
Find the average value ${h}_{ave}$ of the function h on the given interval.
$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),\left[0,\pi \right]$
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The given function h and the interval is,
$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),\left[0,\pi \right]$
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.
${h}_{ave}=\frac{8}{\pi }{\int }_{-1}^{1}{u}^{4}du$
$=\frac{8}{\pi }\left(\frac{2}{5}\right)$
$=\frac{16}{5\pi }$
###### Not exactly what you’re looking for? Drood1980
Consider the function
$h\left(x\right)=8{\mathrm{cos}}^{4}x\mathrm{sin}x$
The average value of the function in the interval $\left[0,\pi \right]$ 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 ${h}_{ave}=\frac{16}{5\pi }$