# 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.
$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),\left[0,\pi \right]$
You can still ask an expert for help

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it

## Expert Answer

Mary Darby
Answered 2021-11-17 Author has 11 answers
The given function h and the interval is,
$h\left(x\right)=8{\mathrm{cos}}^{4}\left(x\right)\mathrm{sin}\left(x\right),\left[0,\pi \right]$
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
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 $\left[0,\pi \right]$ 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 ${h}_{ave}=\frac{16}{5\pi }$
###### Not exactly what you’re looking for?

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it