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 OB. (2/3-2)-3 n2 O C. (3-1)x+ (3-1) D. 2 12

It's unclear what the planes are supposed to be, so I'll take [tex]x=a[/tex] and [tex]x=b[/tex] with [tex]0\le a<b<\pi[/tex].The cross sections are disks with diameter [tex]\csc x-\cot x[/tex], so each disk of thickness [tex]\Delta x[/tex] has a volume of[tex]\dfrac{\pi(\csc x-\cot x)^2}4\Delta x[/tex]Then taking infinitesimally thin disks, we find the solid has a volume of[tex]\displaystyle\frac\pi4\int_a^b(\csc x-\cot x)^2\,\mathrm dx[/tex]Since[tex](\csc x-\cot x)^2=2\csc^2x-2\csc x\cot x-1[/tex]and[tex]\dfrac{\mathrm d(\csc x)}{\mathrm dx}=-\csc x\cot x[/tex][tex]\dfrac{\mathrm d(\cot x)}{\mathrm dx}=-\csc^2x[/tex]it follows that the volume is[tex]\displaystyle\frac\pi4\left(-2\cot x+2\csc x-x\right)\bigg|_a^b[/tex][tex]=\dfrac\pi4(2(\cot a-\cot b)+2(\csc b-\csc a)+a-b)[/tex][tex]=\dfrac\pi4\left(2\tan\dfrac b2-2\tan\dfrac a2+a-b\right)[/tex]