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