PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple Integrals Dr Mervyn Roy (S6)

PA215: Many variables Revision - integrals over a plane Easy when integration directions along co-ordinate axes – limits of integration are constants

PA215: Many variables Integrals over irregular regions of a plane But… no reason why integration directions should be along co-ordinate axes

PA215: Many variables Section 3, Example 1 Evaluate

PA215: Many variables Evaluate Section 3, Example 1

PA215: Many variables Change of order of integration - Need to be careful with the limits! Section 4, Example 1 Evaluate

PA215: Many variables Section 4, Example 2 Evaluate

PA215: Many variables Section 4, Example 2 Evaluate

PA215: Many variables Solid angle - Solid angle defined in analogous manner to ordinary angle - Solid angle (steradians) subtended by a surface is

PA215: Many variables Section 14, Example 1 - Find the solid angle subtended by a sphere

PA215: Many variables Section 15. Example 1 - Look for some symmetry in the function Sketching surfaces

PA215: Many variables Section 15. Example 1 - Look for some symmetry in the function Sketching surfaces

PA215: Many variables Sketching surfaces