1. 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) http://www2.le.ac.uk/departments/physics/people/academic-staff/mr6/

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

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

4. PA215: Many variables Section 3, Example 1 Evaluate

5. PA215: Many variables Evaluate Section 3, Example 1

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

7. PA215: Many variables Section 4, Example 2 Evaluate

8. PA215: Many variables Section 4, Example 2 Evaluate

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

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

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

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

13. PA215: Many variables Sketching surfaces