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Find the average value of the functions on the given interval. Average value of f(x)=x on [5,13] :

Find the average value of the functions on the given interval.\newlineAverage value of f(x)=x f(x)=x on [5,13] [5,13]

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Q. Find the average value of the functions on the given interval.\newlineAverage value of f(x)=x f(x)=x on [5,13] [5,13]
  1. Identify Formula: Identify the formula for the average value of a function on an interval [a,b][a, b].\newlineThe formula is: Average value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx.\newlineHere, f(x)=xf(x) = x, a=5a = 5, and b=13b = 13.
  2. Calculate Integral: Calculate the integral of f(x)=xf(x) = x from 55 to 1313.∫513x dx=[x22]513=(1322)−(522)=(1692)−(252)=1442=72\int_{5}^{13} x \, dx = \left[\frac{x^2}{2}\right]_{5}^{13} = \left(\frac{13^2}{2}\right) - \left(\frac{5^2}{2}\right) = \left(\frac{169}{2}\right) - \left(\frac{25}{2}\right) = \frac{144}{2} = 72.
  3. Plug into Formula: Plug the integral result into the average value formula.\newlineAverage value = (1/(13−5))×72=(1/8)×72=9(1/(13-5)) \times 72 = (1/8) \times 72 = 9.

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