# Hard question

• Feb 9th 2010, 10:33 PM
arze
Hard question
A region of the xy plane 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 the x-axis,
(c) the coordinates of the centroid of this area.

Find Also:
(d)The volume obtained when this area is rotated completely about the x-axis,
(e) The first moment of volume about the y-axis,
(f) The centroid of this volume.

My attempt:
I found (a) correct, its value is 2.
First moment of area about Ox is $\displaystyle Ay\simeq \sum^{\pi}_{x=0} y^2\delta x$
$\displaystyle 2y=\int^{\pi}_0(\frac{1}{2}-\frac{1}{2}\cos 2x)dx$
$\displaystyle 2y=[\frac{x}{2}-\frac{1}{4}\sin 2x]^{\pi}_0$
$\displaystyle 2y=\frac{\pi}{2}$
$\displaystyle y=\frac{\pi}{4}$
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 Oy is $\displaystyle \simeq(\pi y^2 \delta x)x$
$\displaystyle Vx\simeq\sum^{\pi}_{x=0}(\pi y^2 x \delta x)$
$\displaystyle \frac{\pi^2}{2}x=\int^{\pi}_0 \pi \sin^3x dx$
$\displaystyle \frac{\pi^2}{2}x=\pi \int^{\pi}_0\sin x(1-\cos^2x)dx$
$\displaystyle \frac{\pi^2}{2}x=\pi [-\cos x+\frac{1}{3}\cos^3x]^{\pi}_0$
$\displaystyle \frac{\pi^2}{2}x=\pi(\frac{4}{3})$
$\displaystyle x=\frac{8}{3\pi}$
answer is $\displaystyle \frac{\pi^3}{4}$
I can do the last one.

Thanks!
• Feb 10th 2010, 05:11 AM
jiboom
deleted
• Feb 10th 2010, 05:34 AM
Calculus26
For c)

obtains 2y = pi/2 not pi^2/2 as you have x not x^2

so Moment is pi/4

and the the y coordinate is pi/8
• Feb 10th 2010, 05:52 AM
jiboom
the first moment $\displaystyle M_y$ is
$\displaystyle M_y= \int^{\pi}_0(\frac{y^2}{2}) dx$

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

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

so
first moment $\displaystyle M_y= \frac{\pi}{4}$

Y co-ord of centroid is $\displaystyle \frac{M_y}{A}=\frac{\frac{\pi}{4}}{2}=\frac{\pi}{8 }$

for volume:

first moment of volume

$\displaystyle M_y=\int^{\pi}_0 \pi y^2 x dx$

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

which is

$\displaystyle \pi( [\frac{x^2}{4}-\frac{x\sin(2x)}{4}-\frac{\cos(2x)}{8}]^{\pi}_0 )$

$\displaystyle =\pi([\frac{\pi^2}{4}-0-\frac{1}{8}] -$$\displaystyle [-\frac{1}{8}])$

$\displaystyle =\frac{\pi^3}{4}$
• Feb 10th 2010, 10:05 PM
arze
I've got another similar question and this time the borders of the area are y=x^2 and y=4. The parts are about the same.
first i was asked to get the area which i found correctly.
then i am asked to find the first moment of area about the x-axis. so i did this:
$\displaystyle M_x=\int^2_{-2}\frac{(4-x^2)^2}{2}dx=\frac{1}{2}[16x-\frac{8x}{3}+\frac{x^5}{5}]^2_{-2}=\frac{256}{15}$
the answer is $\displaystyle \frac{128}{5}$ as far as i can see my logic is correct and so is the working.
• Feb 13th 2010, 12:46 PM
jiboom
Quote:

Originally Posted by arze
I've got another similar question and this time the borders of the area are y=x^2 and y=4. The parts are about the same.
first i was asked to get the area which i found correctly.
then i am asked to find the first moment of area about the x-axis. so i did this:
$\displaystyle M_x=\int^2_{-2}\frac{(4-x^2)^2}{2}dx=\frac{1}{2}[16x-\frac{8x}{3}+\frac{x^5}{5}]^2_{-2}=\frac{256}{15}$
the answer is $\displaystyle \frac{128}{5}$ as far as i can see my logic is correct and so is the working.

i think you better of integrating between the cure and y -axis here as you have stuff between the curve and x-axis you dont want.
so we want (as we both sides of the y-axis i think the 2 outside integral is ok,it gives the right answer so its promising.

$\displaystyle 2\int_0^4y(f(y))$

$\displaystyle = 2\int_0^4 y(y^{1/2})$

$\displaystyle =2 [\frac{2x^{5/2}}{5}]_0^4$

$\displaystyle =128/5$