Find the average value of the function over the given interval.
-$y = 6 - x ^ { 2 } ; [ - 3,2 ]$
A) $\frac { 109 } { 15 }$
B) $\frac { 11 } { 3 }$
C) $\frac { 1 } { 15 }$
D) $- \frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

The formula for finding the average value of a function f(x) over the interval [a,b] is given by:

$$	ext{Average value of } f(x) = \frac{1}{b-a} \int_{a}^{b} f(x)\, dx$$

Using this formula, we can find the average value of y = 6 - x^2 over the interval [-3,2]:

$$	ext{Average value of } y = \frac{1}{2-(-3)} \int_{-3}^{2} (6-x^2)\, dx$$

$$= \frac{1}{5} \left[6x - \frac{x^3}{3}ight]_{-3}^{2}$$

$$= \frac{1}{5} \left[(12-\frac{8}{3}) - (-18+\frac{54}{3})ight]$$

$$= \frac{1}{5} \cdot \frac{109}{3}$$

$$= \frac{11}{3}$$

Therefore, the best choice is B, which is $\frac{11}{3}$.

Find the Average Value of the Function Over the Given y=6−x2;[−3,2]y = 6 - x ^ { 2 } ; [ - 3,2 ]y=6−x2;[−3,2]

Find the Average Value of the Function Over the Given $y = 6 - x ^ { 2 } ; [ - 3,2 ]$