A region of thexyplane is defined by the inequalities

$\displaystyle 0\leq y\leq \sin x$, $\displaystyle 0\leq x\leq \pi$

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 $\displaystyle Ay\simeq \sum^{\pi}_{x=0} y^2\delta x$

$\displaystyle 2y=\int^{\pi}_0(\frac{1}{2}-\frac{1}{2}\cos 2x)dx$

$\displaystyle 2y=[\frac{x}{2}-\frac{1}{4}\sin 2x]^{\pi}_0$

$\displaystyle 2y=\frac{\pi}{2}$

$\displaystyle y=\frac{\pi}{4}$

This gives me the correct answer, but when i proceed to (c) i have problems. The x-coordinate of the centroid would be $\displaystyle \frac{\pi}{2}$ and the y-coordinate would be $\displaystyle \frac{\pi}{4}$ but the answer is $\displaystyle \frac{\pi}{8}$

I also found (d) correctly it is $\displaystyle \frac{\pi^2}{2}$

Now (e) we have First moment of $\displaystyle \delta V$ about Oyis $\displaystyle \simeq(\pi y^2 \delta x)x$

$\displaystyle Vx\simeq\sum^{\pi}_{x=0}(\pi y^2 x \delta x)$

$\displaystyle \frac{\pi^2}{2}x=\int^{\pi}_0 \pi \sin^3x dx$

$\displaystyle \frac{\pi^2}{2}x=\pi \int^{\pi}_0\sin x(1-\cos^2x)dx$

$\displaystyle \frac{\pi^2}{2}x=\pi [-\cos x+\frac{1}{3}\cos^3x]^{\pi}_0$

$\displaystyle \frac{\pi^2}{2}x=\pi(\frac{4}{3})$

$\displaystyle x=\frac{8}{3\pi}$

answer is $\displaystyle \frac{\pi^3}{4}$

I can do the last one.

Thanks!