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From Math 2220 Class 25 V1cc Double Integral Problems From Math 2220 Class 25 Triple Integrals A Generic Elementary Dr. Allen Back Region Some Basic Triple Integal Setup Problems Oct. 24, 2014 Why Cont. Fcns are Integrable

  1. From Math 2220 Class 25 V1cc Double Integral Problems From Math 2220 Class 25 Triple Integrals A Generic Elementary Dr. Allen Back Region Some Basic Triple Integal Setup Problems Oct. 24, 2014 Why Cont. Fcns are Integrable Double Integrals

  2. Double Integral Problems From Math 2220 Class 25 Use a double (or triple) integral to find the volume bounded by V1cc the four planes Double Integral z = 0 Problems x = 1 Triple Integrals y = 2 A Generic Elementary x + 2 y + 3 z = 6 . Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  3. Triple Integrals From Math 2220 Class 25 V1cc Integrating over an elementary region D leads to triple integrals Double such as Integral Problems � b � d ( x ) � h ( x , y ) Triple Integrals f ( x , y , z ) dz dy dx . A Generic c ( x ) g ( x , y ) a Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  4. Triple Integrals From Math Integrating over an elementary region D leads to triple integrals 2220 Class 25 such as V1cc � b � d ( x ) � h ( x , y ) Double Integral f ( x , y , z ) dz dy dx . Problems a c ( x ) g ( x , y ) Triple Integrals Please note: The inner limits of integration encode the answer A Generic Elementary to the question: Region Some Basic For fixed x and y , what is the intersection of a line Triple Integal Setup x = cst , y = cst parallel to the z -axis with D ? Problems Why Cont. Fcns are Integrable Double Integrals

  5. Triple Integrals From Math Integrating over an elementary region D leads to triple integrals 2220 Class 25 such as V1cc � b � d ( x ) � h ( x , y ) Double Integral f ( x , y , z ) dz dy dx . Problems c ( x ) g ( x , y ) a Triple Integrals � b � d ( x ) Please note: The outer limits c ( x ) describe A Generic a Elementary Region 1 Which lines x = cst , y = cst intersect D . Some Basic 2 Or equivalently what is the projection of D into the Triple Integal Setup xy -plane. Problems Why Cont. Fcns are Integrable Double Integrals

  6. Triple Integrals Integrating over an elementary region D leads to triple integrals From Math 2220 Class 25 such as V1cc � b � d ( x ) � h ( x , y ) Double Integral f ( x , y , z ) dz dy dx . Problems c ( x ) g ( x , y ) a Triple Integrals So on all but the easiest problems you want to A Generic Elementary 1 Sketch some version of D and perhaps indicate the lines Region parallel to the axis of the inner variable of integration. Some Basic Triple Integal 2 Sketch the projection of the region into the plane of the Setup Problems two outer variables. Why Cont. Fcns are Integrable Double Integrals

  7. Triple Integrals From Math 2220 Class 25 V1cc Setting up triple integrals (especially reliably) can be hard! Double Integral One helpful principle that often comes up in the inner setup is Problems the idea that the only places where a graph z = g ( x , y ) can Triple Integrals change from being below a graph z = h ( x , y ) to above are at A Generic points where g ( x , y ) = h ( x , y ) . Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  8. Triple Integrals From Math 2220 Class 25 V1cc Setup the triple integral for the volume of the region bounded Double Integral by the paraboloids Problems Triple z = 4 − x 2 − y 2 and z = 1 + x 2 + y 2 . Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  9. Triple Integrals From Math 2220 Class 25 Setup the triple integral for the volume of the region bounded V1cc by the paraboloids Double Integral z = 4 − x 2 − y 2 2 and z = 1 + x 2 + 3 y 2 Problems 2 . Triple Integrals A Generic (Note the intersection here is much more complicated than the Elementary previous example (no longer a curve lying in a plane), but the Region Some Basic setup is not much more difficult.) Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  10. Triple Integrals From Math 2220 Class 25 V1cc ��� Double xy dV Integral Problems Triple Integrals for the region between the planes x + 2 y − z = 0 and A Generic y − z = 0 and above the triangle with vertices (0 , 0 , 0), Elementary Region (0 , 1 , 0), and (1 , 0 , 0) . Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  11. Triple Integrals From Math 2220 Class 25 V1cc Double Integral The volume of the solid bounded by the parabolic cylinder Problems x = y 2 , the xy -plane, and the plane x + z = 1 . Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  12. Triple Integrals From Math 2220 Class 25 V1cc Double Integral Setup the triple integral of f ( x , y , z ) over the tetrahedron with Problems vertices (0 , 0 , 0) , (3 , 2 , 0) , (0 , 3 , 0) , and (0 , 0 , 2) . Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  13. Triple Integrals From Math 2220 Class 25 V1cc Double Setup the triple integral of f ( x , y , z ) over the tetrahedron with Integral Problems vertices (0 , 0 , 0) , (3 , 2 , 0) , (0 , 3 , 0) , and (0 , 0 , 2) . Triple Integrals How about changing (0 , 0 , 2) to (1 , 1 , 2)? A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  14. Triple Integrals From Math 2220 Class 25 V1cc Double Setup the triple integral of f ( x , y , z ) over he smaller region Integral Problems bounded by the cylinder x 2 + y 2 − 2 y = 0 and the planes Triple x − y = 0 , z = 0 , and z = 3 . Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  15. A Generic Elementary Region From Math 2220 Class 25 V1cc Integrating over an elementary region D leads to triple integrals Double such as Integral Problems � b � d ( x ) � h ( x , y ) Triple Integrals f ( x , y , z ) dz dy dx . A Generic c ( x ) g ( x , y ) a Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  16. A Generic Elementary Region From Math Region in the first octant bounded 2220 Class 25 above by z = x 2 + y 2 + 1 V1cc Double below by z = 0 Integral Problems and on the sides by Triple Integrals the xz plane A Generic the yz plane Elementary Region the plane y = 2 Some Basic Triple Integal parabolic cylinder x = ( y − 1) 2 + 1 . Setup Problems Why Cont. Fcns are Integrable Double Integrals

  17. A Generic Elementary Region From Math the whole solid 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  18. A Generic Elementary Region From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  19. A Generic Elementary Region From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  20. A Generic Elementary Region From Math 2220 Class 25 V1cc Double Integral � 2 � ( y − 1) 2 +1 � 1+ x 2 + y 2 Problems f ( x , y , z ) dzdxdy . Triple y =0 x =0 z =0 Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  21. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  22. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  23. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  24. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  25. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

  26. Some Basic Triple Integal Setup Problems From Math 2220 Class 25 V1cc Double Integral Problems Triple Integrals A Generic Elementary Region Some Basic Triple Integal Setup Problems Why Cont. Fcns are Integrable Double Integrals

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