Question

A solid lies between planes perpendicular to the x-axis at x tox=. The cross sections perpendicular to the x-axis are circular disks with diameters running from the curve y cotx to the curve y csc x.
Find the volume of the solid.
A. +1) OA 6 2 O
B. (2/3-2)-3 n2 O
C. (3-1)x+ (3-1)
D. 2 12

Answer 1

It's unclear what the planes are supposed to be, so I'll take $x=a$ and $x=b$ with $0\le a<b<\pi$.

The cross sections are disks with diameter $\csc x-\cot x$, so each disk of thickness $\Delta x$ has a volume of

$\dfrac{\pi(\csc x-\cot x)^2}4\Delta x$

Then taking infinitesimally thin disks, we find the solid has a volume of

$\displaystyle\frac\pi4\int_a^b(\csc x-\cot x)^2\,\mathrm dx$

Since

$(\csc x-\cot x)^2=2\csc^2x-2\csc x\cot x-1$

and

$\dfrac{\mathrm d(\csc x)}{\mathrm dx}=-\csc x\cot x$

$\dfrac{\mathrm d(\cot x)}{\mathrm dx}=-\csc^2x$

it follows that the volume is

$\displaystyle\frac\pi4\left(-2\cot x+2\csc x-x\right)\bigg|_a^b$

$=\dfrac\pi4(2(\cot a-\cot b)+2(\csc b-\csc a)+a-b)$

$=\dfrac\pi4\left(2\tan\dfrac b2-2\tan\dfrac a2+a-b\right)$