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

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

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Full solution

Q. Find the average value of the functions on the given interval.\newlinea) Avêrage value of f(x)=x f(x)=x on [5,13] [5,13] :
  1. Use Formula: To find the average value of a function f(x)f(x) on the interval [a,b][a, b], use the formula: Average value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx.
  2. Identify Interval: For f(x)=xf(x) = x, the interval is [5,13][5,13]. So a=5a = 5 and b=13b = 13.
  3. Plug Values: Plug the values into the formula: Average value = 1(13−5)\frac{1}{(13-5)} * ∫513x dx\int_{5}^{13} x \, dx.
  4. Calculate Difference: Calculate the difference 13−513 - 5: Average value = (1/8)×∫513x dx(1/8) \times \int_{5}^{13} x \, dx.
  5. Integrate Function: Integrate f(x)=xf(x) = x from 55 to 1313: ∫x dx=12x2\int x \, dx = \frac{1}{2}x^2.
  6. Evaluate Integral: Evaluate the integral from 55 to 1313: (1/2)x2 ∣ from 5 to 13=(1/2)(132)−(1/2)(52)(1/2)x^2 \, | \, \text{from } 5 \text{ to } 13 = (1/2)(13^2) - (1/2)(5^2).
  7. Calculate Squares: Calculate the squares: (12)(169)−(12)(25)(\frac{1}{2})(169) - (\frac{1}{2})(25).
  8. Perform Subtraction: Perform the subtraction: (12)(169)−(12)(25)=(12)(144)(\frac{1}{2})(169) - (\frac{1}{2})(25) = (\frac{1}{2})(144).
  9. Multiply by 11/22: Multiply by 12\frac{1}{2}: (12)(144)=72\left(\frac{1}{2}\right)(144) = 72.
  10. Multiply by Reciprocal: Now multiply by the reciprocal of the interval length 18\frac{1}{8}: Average value = 18×72\frac{1}{8} \times 72.
  11. Calculate Final Result: Calculate the final result: Average value = 99.

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