Find the average value of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval $[ - 2,2 ]$ .
A) $\frac { \pi } { 2 }$
B) $\pi$
C)8 $\pi$
D)3 $\pi$
E)2 $\pi$
F)4 $\pi$
G)16 $\pi$
H)6 $\pi$

Question

Find the average value of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval $[ - 2,2 ]$ .
A) $\frac { \pi } { 2 }$ 
B) $\pi$ 
C)8 $\pi$ 
D)3 $\pi$ 
E)2 $\pi$ 
F)4 $\pi$ 
G)16 $\pi$ 
H)6 $\pi$

Quizplus · Accepted Answer

The average value of a function $f(x)$ over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) dx$. For $f(x) = \sqrt{4 - x^2}$ on $[-2, 2]$, the integral represents the area of a semicircle with radius 2. The area of a full circle with radius 2 is $\pi r^2 = 4\pi$, so the area of the semicircle is $2\pi$. The average value is $\frac{1}{2-(-2)} \cdot 2\pi = \frac{1}{4} \cdot 2\pi = \frac{\pi}{2}$.

Find the Average Value of the Function f(x)=4−x2f ( x ) = \sqrt { 4 - x ^ { 2 } }f(x)=4−x2​

Find the Average Value of the Function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$