8. What is the x-coordinate of the centroid of the closed region
R = {(x,y)
Î R2: 0 £ x £ 1, Ö (1-x2) £ y £ 1} ?
|
A. 1/6 |
B. 1/3 |
C. 2/(3(4- p )) |
|
D. 4/3 p |
E. 6/(4- p ) |
|
Solution: The answer is C
By definition, the x coordinate of the center of mass of the given region is:
Where A is the area of the closed region R and f(x)= 1
³ g(x)= Ö (1-x2). Since 0 £ x £ 1, Ö (1-x2) £ y £ 1, we can say that the area of R is the difference between the area of the square of length 1 and the area of a quarter of a circle with radius 1. That is,
Solving for the x coordinate of the center of mass yields,
