Average Value Theorem

Question

If the average value of the function f on the interval 1≤x≤4 is 8, what is the value of ∫41 (3f(x)+2x)dx ?

Note: that is, the integral from 1 to 4 of 3f(x)+2x)dx

Answer 1

From the linearity of the integral, we can break the problem apart as follows:

∫₁⁴[3f(x) + 2x] dx = 3∫₁⁴ f(x) dx + 2∫₁⁴ x dx

The first integral on the right hand side can be evaluated using the mean value theorem. We know that the average value of the function on 1 < x < 4 is 8, and the interval has length 4 - 1 = 3. The integral must therefore be equal to the average value times the length of the interval, or

∫₁⁴ f(x) dx = 8 * 3 = 24.

The second term in the first equation can be found using the power rule, so that

∫₁⁴ x dx = x²/2 |₁⁴ = 8 - 1/2 = 15/2 = 7.5.

Multiplying by the appropriate coefficients gives

3 * 24 + 2 * 15 / 2 = 72 + 15 = 87.