Passage

The average value of a function f(x) over the interval, [a,b] is the number $$\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$$
The square root $$\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$$ is called the root mean square of f on [a, b]. The average value of $$\displaystyle \mu $$ is attained id f is continuous on [a, b].
Mathematics

The average value of f(x)=$$\displaystyle \frac{\cos ^{2}x}{\sin ^{2}x+4\cos ^{2}x}on\left [ 0,\pi /2 \right ]$$ is


ANSWER

1/6


SOLUTION
$$\cfrac { 1 }{ \pi /2-0 } \int _{ 0 }^{ \pi /2 }{ \cfrac { \cos ^{ 2 }{ x }  }{ \sin ^{ 2 }{ x } +4\cos ^{ 2 }{ x }  } dx } $$
Substituting $$t=\tan { x } $$
$$=\cfrac { 2 }{ \pi  } \int _{ 0 }^{ \infty  }{ \cfrac { dt }{ \left( { t }^{ 2 }+4 \right) \left( { t }^{ 2 }+1 \right)  }  } \\ =\cfrac { 2 }{ \pi  } \cfrac { 1 }{ 3 } \int _{ 0 }^{ \infty  }{ \left( \cfrac { 1 }{ { t }^{ 2 }+1 } -\cfrac { 1 }{ { t }^{ 2 }+4 }  \right) dt } \\ =\cfrac { 1 }{ 6 } $$
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Single Correct Hard Published on 17th 09, 2020
Mathematics

The average value of the pressure varying from 2 to 10 atm if the pressure p and the volume v are related by $$\displaystyle pv^{3/2}=160$$ is


ANSWER

$$\displaystyle \frac{40}{3\sqrt{20}\left ( 3\sqrt{10}+3\sqrt{2} \right )}$$


SOLUTION
As p varies from 2 to 10 atm, v transverses the interval $$\left[ 4\sqrt [ 3 ]{ 4 } ,4\sqrt [ 3 ]{ 100 }  \right] $$
Average value =$$=\cfrac { 1 }{ 4\left( \sqrt [ 3 ]{ 100 } -\sqrt [ 3 ]{ 4 }  \right)  } \int _{ 4\sqrt [ 3 ]{ 4 }  }^{ 4\sqrt [ 3 ]{ 100 }  }{ 160{ v }^{ -3/2 }dv } \\ =\cfrac { 40 }{ \sqrt [ 3 ]{ 20 } \left( \sqrt [ 3 ]{ 10 } +\sqrt [ 3 ]{ 2 }  \right)  } $$
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Single Correct Medium Published on 17th 09, 2020
Mathematics

The average ordinate of y= sin x over the interval $$\displaystyle \left [ 0,\pi  \right ]$$ is


ANSWER

$$\displaystyle 2/\pi $$


SOLUTION
$$\mu =\cfrac { 1 }{ x } \int _{ 0 }^{ \pi  }{ \sin { x } dx } =-\cfrac { 1 }{ x } { \left[ \cos { x }  \right]  }_{ 0 }^{ \pi  }=\cfrac { 2 }{ \pi  } $$
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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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