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lecture28-09 - 18.02 Multivariable Calculus(Spring 2009...

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18.02 Multivariable Calculus (Spring 2009): Lecture 28 Triple Integrals. Cylindrical Coordinates April 16 Reading Material: From Simmons : 20.5 and 20.6. Last time: Simply Connected Regions. Today: Triple Integrals. Cylindrical Coordinates. 2 What we did in 2D and what we need to do in 3D 2D 3D Double integrals Triple integrals Polar coordinates Cylindrical & spherical coordinates 2D Applications 3D Applications Line integrals Line integrals in 3D Surface integrals Green’s Theorem for flux Gauss (divergence) Theorem Green’s Theorem for work Stoke’s Theorem 3 Introduction to triple integrals The 2D case: We recall that given a function f ( x, y ) and a region R 1
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we can write R f ( x, y ) dA = b a g ( x ) h ( x ) f ( x, y ) dy dx. To determine the limits of the outer variable we look at the shadow of R on the axis relative to the outer variable. We also recall that R 1 dA = Area( R ) . The 3D case: Here usually we have a function f ( x, y, z ) and a 3D region D We want to define D f ( x, y, z ) dV where dV denotes the infinitesimal element of volume. The definition of the integral above should
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