Triple Integrals over General Regions

Calculus 3 · Axiom Academy

LESSON Triple Integrals over General Regions To integrate over a solid, project it onto a plane and let the last variable run between two surfaces — the inner limits become functions of the outer ones. 1. Type I: Project onto the xy-plane A solid is Type I (z-simple) when a vertical line through its shadow enters through one surface and leaves through another. Project E straight down onto the xy -plane to get the region D . Then for each (x,y) in D , the height z runs from the bottom surface z=u_1(x,y) up to the top surface z=u_2(x,y) . Integrate z first (inner); then the shadow D (outer double integral) 2. Type II: Project onto the xz-plane The same solid is Type II (y-simple) when we instead project onto the xz -plane. For each point (x,z) in that shadow D , the variable y runs from a left surface y=u_1(x,z) to a right surface y=u_2(x,z) — now the fibers point in the y -direction. Integrate y first (inner); then the shadow D in the xz -plane 3. Type III: Project onto the yz-plane Finally, the solid is Type III (x-simple) when we project onto the yz -plane. For each point (y,z) in that shadow D , the variable x runs from a back surface x=u_1(y,z) to a front surface x=u_2(y,z) — the fibers now point in the x -direction. Integrate x first (inner); then the shadow D in the yz -plane

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