Find the exact average value of the function over the given interval. f(x) = 8x + 12 over [1, 7]

Question

Find the exact average value of the function over the given interval.
f(x) = 8x + 12 over [1, 7]

 

So I plugged in 7 and 1 and got 68 and 20. I then proceeded to subtract them using the fundamental theorem. I go 48 I then divided by 6 (7-1) to get the average but that didn't work.

 

So then I took a different approach and used 48 as the height of a triangle and multiplied by half the base (3) and then divided by six but that is wrong as well. How do I do this problem.

Answer 1

The average value of f(x) over the interval [a,b] is given by

 

<f> = [∫_a^b f(x) dx] / [∫_a^b dx]

 

<f> = [∫₁⁷ (8x + 12) dx] / ∫₁⁷ dx

 

The numerator gives:

 

[4(7²) + 12(7)] - [4(1²) + 12(1)] = 264

 

The denominator gives:

 

7 - 1 = 6

 

<f> = 264/6 = 44

Answer 2

Take the integral of f(x).

 

F(x) = ∫f(x)dx

 

 

Then evaluate   F(7) - F(1)