HSC Extension 1 and 2 Mathematics/Integration

AreaEdit

  • Fundamental Theorem of Calculus:  , where  

Area between two curvesEdit

Volume of solids of revolutionEdit

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 integrationEdit

Trapezoidal ruleEdit

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

Simpson's ruleEdit