HSC Extension 1 and 2 Mathematics/Integration

Area 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. . Approximate integration Trapezoidal rule One interval (2 function values): -intervals ( function values): Simpson's rule