Volumes with Known Cross Sections
Not all volume problems involve revolution. Some have cross sections that are squares, triangles, or semicircles.
Formula: V = ∫ from a to b A(x) dx
where A(x) is the area of the cross section at position x.
Common cross sections:
Square: A = s² where s = f(x) - g(x)
Equilateral triangle: A = (√3/4)s²
Semicircle: A = (π/8)s²
Rectangle with height 3: A = 3s
Setup: The base is the region between two curves. Cross sections are perpendicular to the x-axis (or y-axis).
AP exam: This appears on free response regularly. The setup is the hard part — once you write the integral correctly, the computation is straightforward.
Reference:
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