Step-by-step explanation:
Suppose we have a curve, y = f(x).
y = f(x)
x = a x = b
Imagine that the part of the curve between the ordinates x = a and x = b is rotated about the
x-axis through 360◦
. The curve would then map out the surface of a solid as it rotated. Such
solids are called solids of revolution. Thus if the curve was a circle, we would obtain the surface
of a sphere. If the curve was a straight line through the origin, we would obtain the surface of
a cone. Now we already know what the formulae for the volumes of a sphere and a cone are,
but where did they come from? How can they calculated? If we could find a general method
for calculating the volumes of the solids of revolution then we would be able to calculate, for
example, the volume of a sphere and the volume of a cone, as well as the volumes of more
complex solids.
To see how to carry out these calculations we look first at the curve, together with the solid it
maps out when rotated through 360◦
.
y = f(x)
Now if we take a cross-section of the solid, parallel to the y-axis, this cross-section will be a
circle. But rather than take a cross-section, let us take a thin disc of thickness δx, with the face
of the disc nearest the y-axis at a distance x from the origin.
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