Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .
A)Average = 5
B) Average = 6
C) Average = 12
D) Average = 10
E) Average = 7

Question

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .  
A)Average = 5
B) Average = 6
C) Average = 12
D) Average = 10
E) Average = 7

Quizplus · Accepted Answer

The average value of a function $f(x)$ over the interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) dx$. For $f(x) = 9x^2 - 6x$ over $[0, 2]$, the average value is calculated as follows:$$\text{Average} = \frac{1}{2-0} \int_{0}^{2} (9x^2 - 6x) dx = \frac{1}{2} \left[ 3x^3 - 3x^2 \right]_0^2 = \frac{1}{2} \left[ (3 \cdot 2^3 - 3 \cdot 2^2) - (0) \right] = \frac{1}{2} \left[ 24 - 12 \right] = \frac{12}{2} = 6.$$

Find the Average of the Function f(x)=9x2−6xf ( x ) = 9 x ^ { 2 } - 6 xf(x)=9x2−6x

Find the Average of the Function $f ( x ) = 9 x ^ { 2 } - 6 x$