A region of thexyplane is defined by the inequalities

,

Find:

(a) the area of the region

(b) The first moment of area about thex-axis,

(c) the coordinates of the centroid of this area.

Find Also:

(d)The volume obtained when this area is rotated completely about thex-axis,

(e) The first moment of volume about they-axis,

(f) The centroid of this volume.

My attempt:

I found (a) correct, its value is 2.

First moment of area about Oxis

This gives me the correct answer, but when i proceed to (c) i have problems. The x-coordinate of the centroid would be and the y-coordinate would be but the answer is

I also found (d) correctly it is

Now (e) we have First moment of about Oyis

answer is

I can do the last one.

Thanks!