By Nicholas Lawton (P860) on Thursday, April 8, 1999 - 02:37 pm:
In a diagram, the shaded region R, is bounded by the curve with
equatiion y = e2x, 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 & p the volume of
the solid generated.
[Here's a picture. - The Editor]
By Richard Samworth (Rjs57) on Saturday, April 10, 1999 - 01:40 pm:
Dear Nicholas,
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)pe4x,
since the volume of a cylinder is pr2h.
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
|
Hope this helps.
Richard
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