Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the volume of the solid generated when R is revolved about the x-axis?

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Answer 1 · 2016-05-16T14:59:33

#pi/2# cubic units.

The region extends from y-axis to the common point #(pi/4, 1/sqrt 2).

The volume is the difference between the volumes, for the separate

curves = #pi int# (difference between y^2 values) dx,

between .#x = 0 and x = pi/4#,

So, the volume is #pi int (cos^2x - sin^2x) dx#, from #x = 0 to x = pi/4#

#= pi int cos 2x dx# from #x = 0 to x = pi/4#

#=pi[ (sin 2x )/ 2 ],# between .#x = 0 and x = pi/4#

#pi(sin (pi/2)-0)/2=pi/2#, cubic units.