In a diagram, the shaded region R, is bounded by the curve with equatiion y = e^2x , the line x=1/2 & the co-ordinate axes. The region R is rotated through 360 degrees about the x axis. find in terms of e & pi the volume of the solid generated.

[Here's a picture. - The Editor]

What you need to do here is to slice your shape into discs with lots of planes perpendicular to the x-axis, so that each disc has width dx. The volume of a disc with left-hand end-point x, and right-hand end-point x+dx is approximately (dx)pe^4x, since the volume of a cylinder is pr² h.

Now, in order to calculate the total volume, we need to add up the volumes of these discs; we turn this finite sum into an integral by letting dx tend to zero. If this doesn't make much sense, think of approximating the area under a curve by rectangles of width dx and letting dx tend to zero: we end up with an integral.

Thus the total volume is

ó õ 1/2 0  pe4x dx

which I will leave you to calculate.

Hope this helps.

Richard