Download Triple Integrals - Engineering Mathematics and more Lecture notes Engineering Mathematics in PDF only on Docsity! MATH147 CALCULUS 2 TRIPLE INTEGRALS Change of Variables Sometimes the evaluation of an iterated integral can be simplified by reversing the order of integration. Illustration: The integral, given above, cannot be evaluated by performing the x-integration first since there is no elementary antiderivative, let and of which is 2xdx. Evaluate this integral by expressing it as an equivalent iterated integral with the order of integration reversed. Exercises 2 Evaluate the integral by first reversing the order of integration. 1. Ans: 2. Ans: Evaluation of Triple Integrals Over Rectangular Boxes A double integral can be evaluated by two successive single iterations. A triple integral can be evaluated by three successive iterations Fubini’s Theorem Let G be the rectangular box defined by the inequalities If is continuous on the region G, then Triple Integral Notations: 3! Possible orders of integration: f(x,y,z) b a xy xy yxz yxz dzdydxzyxf )( )( ),( ),( 2 1 2 1 ),,( f e zy zy zyx zyx dxdydzzyxf )( )( ),( ),( 2 1 2 1 ),,( d c yx yx yxz yxz dzdxdyzyxf )( )( ),( ),( 2 1 2 1 ),,( b a xz xz zxy zxy dydzdxzyxf )( )( ),( ),( 2 1 2 1 ),,( f e zx zx zxy zxy dydxdzzyxf )( )( ),( ),( 2 1 2 1 ),,( d c yz yz zyx zyx dxdzdyzyxf )( )( ),( ),( 2 1 2 1 ),,( Exercises 3
> Evaluate the iterated integral
1. Ans: 16
Recommended Readings: Calculus Early Transcendentals (Wiley Custom Edition 10ed by Anton) 1. Read pages 1039 – 1045 2. Read examples 2, 3, 4 and 5 pages 1042 – 1045 Application: Area of Rectangular Regions by Double Integration Example: Use double integration to find the area of the plane region enclosed by the given curves. 1. Bounded above by , above by , right by and left by Solution: Graph the plane region, using a vertical strip and from Fubini’s Theorem, A = A = A = A = A = sq.u (x, y) x = 0 (x, 0) x = 1 x
