From Math 2220 Class 27 V1 Double Integral Problems From Math 2220 Class 27 Triple Integrals Some Basic Triple Integal Dr. Allen Back Setup Problems Change of Coordinates Polar/Sph/Cyl Oct. 29, 2014 Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Double Integral Problems From Math 2220 Class 27 Use a double (or triple) integral to find the volume bounded by V1 the four planes Double Integral z = 0 Problems x = 1 Triple Integrals y = 2 Some Basic Triple Integal x + 2 y + 3 z = 6 . Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 . Some Basic c ( x ) g ( x , y ) a Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math Integrating over an elementary region D leads to triple integrals 2220 Class 27 such as V1 � 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 Some Basic Triple Integal to the question: Setup Problems For fixed x and y , what is the intersection of a line Change of Coordinates x = cst , y = cst parallel to the z -axis with D ? Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math Integrating over an elementary region D leads to triple integrals 2220 Class 27 such as V1 � 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 Some Basic a Triple Integal Setup 1 Which lines x = cst , y = cst intersect D . Problems 2 Or equivalently what is the projection of D into the Change of Coordinates xy -plane. Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals Integrating over an elementary region D leads to triple integrals From Math 2220 Class 27 such as V1 � 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 Some Basic Triple Integal 1 Sketch some version of D and perhaps indicate the lines Setup Problems parallel to the axis of the inner variable of integration. Change of 2 Sketch the projection of the region into the plane of the Coordinates two outer variables. Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 Some Basic points where g ( x , y ) = h ( x , y ) . Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 Setup the triple integral for the volume of the region bounded V1 by the paraboloids Double Integral z = 4 − x 2 − y 2 2 and z = 1 + x 2 + 3 y 2 Problems 2 . Triple Integrals Some Basic (Note the intersection here is much more complicated than the Triple Integal previous example (no longer a curve lying in a plane), but the Setup Problems setup is not much more difficult.) Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 ��� Double xy dV Integral Problems Triple Integrals for the region between the planes x + 2 y − z = 0 and Some Basic y − z = 0 and above the triangle with vertices (0 , 0 , 0), Triple Integal Setup (0 , 1 , 0), and (1 , 0 , 0) . Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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)? Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Triple Integrals From Math 2220 Class 27 V1 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 Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Some Basic Triple Integal Setup Problems From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Change of Coordinates From Math ∆ A ∼ r ∆ r Delta θ by geometry 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Change of Coordinates From Math The Polar Coordinate Transformation 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
Change of Coordinates From Math � � � � x u x v 2220 Class 27 � � � � ∆ A ∼ � ∆ u ∆ v � � � � y u y v V1 � � � Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals
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