B. VOLUME
B1. Solids with Known Cross Sections
If the area of a cross section of a solid is known and can be expressed in terms of x, then the volume
of a typical slice, ΔV, can be determined. The volume of the solid is obtained, as usual, by letting the
number of slices increase indefinitely. In Figure N7–9, the slices are taken perpendicular to the x-axis
so that ΔV = A(x) Δx, where A(x) is the area of a cross section and Δx is the thickness of the slice.
FIGURE N7–9
EXAMPLE 5
A solid has as its base the circle x^2 + y^2 = 9, and all cross sections parallel to the y-axis are squares.
Find the volume of the solid.
SOLUTION:FIGURE N7–10
In Figure N7–10 the element of volume is a square prism with sides of length 2y and thickness Δx,
so