Question

Let ( D ) be the region of the plane inside the unit circle, above the line ( x=y ), and to the right of the ( Y )-axis. What is the average ( X ) coordinate of a point in this region?

Answer 1

To find the average X coordinate of the given region, calculate the average of all the X coordinates of points in that region, which is 0.5.

Step-by-step explanation:

To find the average X coordinate of a point in the given region, we need to calculate the average of all the X coordinates of points in that region. The region is defined as inside the unit circle, above the line x=y, and to the right of the Y-axis.

In this region, the X coordinates range from 0 to 1. To calculate the average, we integrate the X coordinate over this range and divide by the length of the range:

Using the integral, the average X coordinate is 1/2, or 0.5.

Answer 2

The average X coordinate of a point in the given region is 0.5.

Step-by-step explanation:

To find the average X coordinate of a point in the region described, we need to find the average distance of all points in the region from the Y-axis. Since the region lies inside the unit circle, above the line x=y, and to the right of the Y-axis, the X coordinate of each point in the region will be positive. Due to the symmetry of the unit circle, we can conclude that the average X coordinate of a point in this region is 0.5.