8.3 Volumes
(I) Volumes of solids with known cross sections
If we know the formula for the region of a cross section, we can find the volume of the solid having this cross section with the help of the definite integral.
(1) If the cross section is perpendicular to the xaxis and it’s area is a function of x, say A(x), then the volume (v) of the solid on [ a, b ] is given by v =
(2) If the cross section is perpendicular to the yaxis and its area is a function of y, say A(y), then the volume (v) of the solid on [ a, b ] is given by v =
Example 13 Consider a solid whose base is the region inside the circle x^{2} + y^{2} = 4. If cross sections taken perpendicular to the xaxis are squares. Find the volume of this solid.
Solution : Let A be the area of an arbitrary square cross section which is perpendicular to the xaxis
Therefore the volume of the solid
Example 14 A solid has its base is the region bounded by the lines x + 2y = 6, x = 0 and y = 0 and the cross sections taken perpendicular to xaxis are circles. Find the volume the solid.
Solution : The cross section is a circle, perpendicular to the xaxis. Naturally the diameter of this circular cross section has its ends on the xaxis and the line x + 2y = 6, which the function of x.
\ The volume (v) of the solid is
Example 15 A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the xaxis are equilateral triangles. Find its volume.
Solution : The cross section is an equilateral triangle, perpendicular to the xaxis. Its side has its ends on the line x + y = 4 and the xaxis.
\ Side (s) = 4  x
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