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Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces

Multiple Integrals. Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15. Dr Mervyn Roy (S6) http:// www2.le.ac.uk/departments/physics/people/academic-staff/mr6 /. Revision - integrals over a plane.

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Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces

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  1. Multiple Integrals • Integrals over irregular regions • Change of order of integration • Solid angle • Sketching surfaces • - Chapter 10, section 3,4,14,15 Dr Mervyn Roy (S6) http://www2.le.ac.uk/departments/physics/people/academic-staff/mr6/

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

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

  4. Section 3, Example 1 Evaluate whereis the region bounded by

  5. Section 3, Example 1 Evaluate whereis the region bounded by

  6. Change of order of integration - Need to be careful with the limits! Section 4, Example 1 whereis the region bounded by Evaluate

  7. Section 4, Example 2 Evaluate

  8. Section 4, Example 2 Evaluate

  9. 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

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

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

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

  13. Sketching surfaces

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