HSC Extension 1 and 2 Mathematics/Integration

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  • Fundamental Theorem of Calculus: , where

Area between two curves[edit]

Volume of solids of revolution[edit]

Recall that the volume of a solid can be found by where is the cross-sectional area and is the depth of the solid, which is perpendicular to the cross-sectional area.

Similarly, the volume of solids with circular cross sections can be calculated by

  • rotating a curve about an axis (generally or axis)
  • integrating to sum the areas of the slices of circles

Since the area of a circle is , then the integral to evaluate the volume of a solid generated by revolving it around the -axis is

Notice this is a sum of areas of the "slices" of circular cross sections of the solid, i.e. .

Approximate integration[edit]

Trapezoidal rule[edit]

  • One interval (2 function values):
  • -intervals ( function values):

Simpson's rule[edit]