Let be an invertible function such that and . Let be the region enclosed by the graphs of and over the interval . The centroid of lies on
A) the line .
B) the line .
C) the line .
D) It depends on the function .
E) none of the above.

Question

Let  be an invertible function such that  and  . Let  be the region enclosed by the graphs of  and  over the interval  . The centroid of  lies on
A) the line . 
B) the line . 
C) the line . 
D) It depends on the function . 
E) none of the above.

Quizplus · Accepted Answer

By symmetry, the centroid of the region bounded by the two curves over a given interval is on the vertical line that passes through the midpoint of the interval. In this case, the interval is [-π/2,π/2], and the midpoint is 0. Therefore, the centroid is on the vertical line x=0, which is the same as the line y-axis. Choice C is the line y-axis, so it is the correct answer. Note that the specific function used to define the region does not matter for this question.

Let Be an Invertible Function Such That and