HSC Extension 1 and 2 Mathematics/Integration
- Fundamental Theorem of Calculus: , where
Area between two curves
Volume of solids of revolution
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. .
- One interval (2 function values):
- -intervals ( function values):