PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle...
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Transcript of PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle...
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/1.jpg)
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/
![Page 2: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/2.jpg)
PA215: Many variables
Revision- integrals over a plane
Easy when integration directions along co-ordinate axes – limits of integration are constants
![Page 3: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/3.jpg)
PA215: Many variables
Integrals over irregular regions of a plane
But… no reason why integration directions should be along co-ordinate axes
![Page 4: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/4.jpg)
PA215: Many variables
Section 3, Example 1
Evaluate where is the region bounded by
![Page 5: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/5.jpg)
PA215: Many variables
Evaluate
Section 3, Example 1
where is the region bounded by
![Page 6: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/6.jpg)
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
![Page 7: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/7.jpg)
PA215: Many variables
Section 4, Example 2
Evaluate
![Page 8: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/8.jpg)
PA215: Many variables
Section 4, Example 2
Evaluate
![Page 9: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/9.jpg)
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
![Page 10: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/10.jpg)
PA215: Many variables
Section 14, Example 1
- Find the solid angle subtended by a sphere
![Page 11: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/11.jpg)
PA215: Many variables
Section 15. Example 1
- Look for some symmetry in the function
Sketching surfaces
![Page 12: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/12.jpg)
PA215: Many variables
Section 15. Example 1
- Look for some symmetry in the function
Sketching surfaces
![Page 13: PA215: Many variables Integrals over irregular regions Change of order of integration Solid angle Sketching surfaces - Chapter 10, section 3,4,14,15 Multiple.](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d385503460f94a11a78/html5/thumbnails/13.jpg)
PA215: Many variables
Sketching surfaces