What is the average value of a function #sec^2x# on the interval #[pi/6, pi/4]#?

Answer 1

#(12-4sqrt3)/pi#

The average value of the function #f(x)# on the interval #[a,b]# is equivalent to
#1/(b-a)int_a^bf(x)dx#
Thus, the average value of #f(x)=sec^2x# on #[pi/6,pi/4]# is
#1/(pi/4-pi/6)int_(pi/6)^(pi/4)sec^2xdx#
Note that #intsec^2xdx=tanx+C#, since the derivative of #tanx# is #sec^2x#.
Also, note that #1/(pi/4-pi/6)=1/(pi/12)=12/pi#.

The average value then equals:

#1/(pi/4-pi/6)int_(pi/6)^(pi/4)sec^2xdx=12/pi[tanx]_(pi/6)^(pi/4)#
#=12/pi[tan(pi/4)-tan(pi/6)]=12/pi[1-sqrt3/3]=12/pi[(3-sqrt3)/3]#
#=4/pi[3-sqrt3]=(12-4sqrt3)/pi#
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Answer 2

To find the average value of the function ( \sec^2(x) ) on the interval ( \left[ \frac{\pi}{6}, \frac{\pi}{4} \right] ), follow these steps:

  1. Evaluate the integral of ( \sec^2(x) ) over the given interval.
  2. Divide the result from step 1 by the length of the interval.

Here are the steps in detail:

  1. Integrate ( \sec^2(x) ) over the interval ( \left[ \frac{\pi}{6}, \frac{\pi}{4} \right] ): [ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec^2(x) , dx ]

Using the integral of ( \sec^2(x) ), which is ( \tan(x) + C ), evaluate the integral at the upper and lower limits of integration: [ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec^2(x) , dx = \tan\left(\frac{\pi}{4}\right) - \tan\left(\frac{\pi}{6}\right) ]

  1. Calculate the length of the interval: [ \text{Length} = \frac{\pi}{4} - \frac{\pi}{6} = \frac{\pi}{12} ]

  2. Compute the average value of ( \sec^2(x) ) over the interval by dividing the result from step 1 by the length of the interval: [ \text{Average value} = \frac{\tan\left(\frac{\pi}{4}\right) - \tan\left(\frac{\pi}{6}\right)}{\frac{\pi}{12}} ]

[ \text{Average value} = \frac{1 - \frac{\sqrt{3}}{3}}{\frac{\pi}{12}} ]

[ \text{Average value} = \frac{12}{\pi} \left(1 - \frac{\sqrt{3}}{3}\right) ]

[ \text{Average value} \approx \frac{12}{\pi} \left(1 - \frac{1.732}{3}\right) ]

[ \boxed{\text{Average value} \approx \frac{12}{\pi} \left(1 - 0.577\right) \approx \frac{12}{\pi} \times 0.423 \approx \frac{50.76}{\pi}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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