Solve the problem.
-Find the average distance from a point $P ( x , y )$ in the first two quadrants of the disk $x ^ { 2 } + y ^ { 2 } \leq 16$ to the origin.
A) $\frac { 4 } { 3 }$
B) 2
C) 1
D) $\frac { 8 } { 3 }$

Question

Quizplus · Accepted Answer

The average distance from a point $P(x, y)$ in the first two quadrants of the disk $x^2 + y^2 \leq 16$ to the origin can be found by integrating the distance formula over the area of interest and then dividing by the area of that region. The distance from any point $P(x, y)$ to the origin is given by $\sqrt{x^2 + y^2}$. To find the average distance, we integrate this over the first two quadrants of the disk and then divide by the area of the half-disk.The area of the half-disk is $\frac{1}{2} \pi r^2$, where $r = 4$, so the area is $8\pi$.The integral setup for the average distance $D$ is:$$D = \frac{1}{8\pi} \int_{0}^{4} \int_{0}^{\sqrt{16-x^2}} \sqrt{x^2 + y^2} \, dy \, dx$$Solving this integral gives us the average distance, which is $\frac{8}{3}$. This calculation involves using polar coordinates to simplify the integration, recognizing that the distance from any point to the origin in polar coordinates is simply the radius $r$, and then integrating $r$ from 0 to 4 over the half-circle area.

Solve the Problem P(x,y)P ( x , y )P(x,y) In the First Two Quadrants of the Disk

Solve the Problem $P ( x , y )$ In the First Two Quadrants of the Disk