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Unformatted text preview: 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 Greens Theorem for flux Gauss (divergence) Theorem Greens Theorem for work Stokes Theorem 3 Introduction to triple integrals The 2D case: We recall that given a function f ( x, y ) and a region R 1 we can write ZZ R f ( x, y ) dA = Z b a " Z 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 ZZ R 1 dA = Area( R ) ....
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