Question

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?

Answer 1

`pi/2` cubic units.

Explanation:

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.