MATH 251 Fall 2023 Section 15 6 Triple integrals Integration over a rectangular box the triple integral tB f x y z dV where B is the rectangular box x y z a x b c y d r z s is By Fubini s theorem if f is continuous on the box then the six following integrals are equal f x y z dxdydz a r Z d Z s c Z b f x y z dydzdx Z b a Z d c Z s c Z b f x y z dxdzdy Z d a Z s f x y z dxdydz Z s r Z b a Z d r r c c Z b a Z s r Z d c Z s Z d r Z b Z s r Z d c Z b a a f x y z dzdydx f x y z dzdxdy f x y z dydxdz Example 1 Evaluate the triple integral tB xy z dV where B 0 2 1 1 2 5 1 2 0 R 2 Example 2 Evaluate the iterated integralR 1 z R 3py 0 3x2ydxdydz Integration over a general solid region 1 Type 1 solid region a region is of type 1 if it lies between the graphs of two continuous functions of x and y that is E x y z x y 2 D u1 x y z u2 x y 3 For type 1 D is the projection of E onto the xy plane f x y z dV x y E D Z u2 x y u1 x y f x y z dz dA 2 Type 2 solid region a region is of type 2 if it lies between the graphs of two continuous functions of y and z that is E x y z y z 2 D u1 y z x u2 y z For type 2 D is the projection of E onto the yz plane f x y z dV x y E D Z u2 y z u1 y z f x y z dx dA 3 Type 3 solid region a region is of type 3 if it lies between the graphs of two continuous functions of x and z that is E x y z x z 2 D u1 x z y u2 x z For type 3 D is the projection of E onto the xz plane f x y z dV x y E D Z u2 x z u1 x z f x y z dy dA Example 3 Evaluate tE zdV where E is the solid tetrahedron bounded by the four planes x 0 y 0 z 0 and x y z 2 4 Example 4 Evaluate tE x2dV where E is the solid that lies below the plane x 2y 3z 6 and above the region on the xz plane bounded by the curves z px z 1 and x 0 5 6 plane y 4 Example 5 Evaluate tE xdV where E is the region bounded by the paraboloid y x2 z2 and the 0 R 2 Example 6 The gure below shows the region of integration for the integralR 8 3pxR 2 y 0 7 f x y z dzdydx Rewrite this integral in ve di erent orders 8 Note that the volume of a solid E is equal to y E 1dV Example 7 Set up but do not evaluate the volume of the tetrahedron with vertices 0 0 0 0 1 1 1 1 0 and 0 1 0 a Write a double integral that gives the volume of the solid with dA dxdz b Write a triple integral that gives the volume of the solid in the order dV dydxdz Both integrals compute the volume of the same solid so they must be equal Do the integrals you found in part a and part b equal
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