Blog: `f(t) = e^sin(t) cos(t), [0, pi/2]` Find the average value of the function on the given interval.

Sunday, July 21, 2013

$\displaystyle{f{{\left({t}\right)}}}={{e}}^{{\sin{{\left({t}\right)}}}}{\cos{{\left({t}\right)}}},{\left[{0},\frac{\pi}{{2}}\right]}$ Find the average value of the function on the given interval.

The average value is the integral of a function over an interval divided by the length of this interval. The length of interval is $\displaystyle\frac{\pi}{{2}}$ here, to find the integral start from the indefinite integral.

Make the substitution $\displaystyle{u}={\sin{{\left({t}\right)}}},$ then $\displaystyle{d}{u}={\cos{{\left({t}\right)}}}{\left.{d}{t}\right.}$ and the integral is

$\displaystyle\int{{e}}^{{u}}{d}{u}={{e}}^{{u}}+{C}={{e}}^{{{\sin{{\left({t}\right)}}}}}+{C}.$

So the definite integral is

$\displaystyle{{e}}^{{{\sin{{\left({t}\right)}}}}}{{\mid}_{{{t}={0}}}^{{\frac{\pi}{{2}}}}}={{e}}^{{1}}-{{e}}^{{0}}={e}-{1}.$

And the average is $\displaystyle\frac{{{2}{\left({e}-{1}\right)}}}{\pi}.$

Make the substitution $\displaystyle{u}={\sin{{\left({t}\right)}}},$ then $\displaystyle{d}{u}={\cos{{\left({t}\right)}}}{\left.{d}{t}\right.}$ and the integral is

$\displaystyle\int{{e}}^{{u}}{d}{u}={{e}}^{{u}}+{C}={{e}}^{{{\sin{{\left({t}\right)}}}}}+{C}.$

So the definite integral is

$\displaystyle{{e}}^{{{\sin{{\left({t}\right)}}}}}{{\mid}_{{{t}={0}}}^{{\frac{\pi}{{2}}}}}={{e}}^{{1}}-{{e}}^{{0}}={e}-{1}.$

And the average is $\displaystyle\frac{{{2}{\left({e}-{1}\right)}}}{\pi}.$

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Sunday, July 21, 2013

$\displaystyle{f{{\left({t}\right)}}}={{e}}^{{\sin{{\left({t}\right)}}}}{\cos{{\left({t}\right)}}},{\left[{0},\frac{\pi}{{2}}\right]}$ Find the average value of the function on the given interval.

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Sunday, July 21, 2013

Find the average value of the function on the given interval.

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What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

$\displaystyle{f{{\left({t}\right)}}}={{e}}^{{\sin{{\left({t}\right)}}}}{\cos{{\left({t}\right)}}},{\left[{0},\frac{\pi}{{2}}\right]}$ Find the average value of the function on the given interval.

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?