Question

How to find the volume of the solid generated by the rotation about the x-axis of the part of the curve of y=sin x, between x=0 and x=pi ?

Answer 1

To use the method of slicing we want to use Volume = (Area of cross-section)(Thickness) Since the curve is being rotated about a line parallel to the x-axis (indeed, it is being rotated about the actual x-axis) Thickness = dx.
The cross-sections are circles. For a circle A = (pi)r^2
r will the the difference between the top value of y and the bottom value of y.
The top curve is y = sinx.
The bottom curve is y = 0 (the x-axis),
so r = sinx - 0 = sinx
So, dV = (pi)(sinx)^2 dx and all you need to do is to integrate that from x = 0 to x = pi.
[You should note that sinx is non-negative for those values of x, although it won't make a difference here.]