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/
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 where is the region bounded by
PA215: Many variables
Evaluate
Section 3, Example 1
where is the region bounded by
PA215: Many variables
Change of order of integration
- Need to be careful with the limits!
Section 4, Example 1
Evaluate where is the region bounded by
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 at a point by an element of area , distance fromis
- 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