Spring 2005 Math 152 Overview Vector Multiple Integrals c 2005 Art Belmonte Thu 03 Mar from a to b is defined by Z b r t dt a Summary Looking ahead to Calc 3 we ll briefly examine vector integrals multiple integrals and vector multple integrals While the limitof a Riemann sum approach is mentioned in practice these types of integrals are evaluated in a mechanistic fashion They are quite easy to do with a TI 89 calculator or in MATLAB This is what we ll usually do especially in Calc 3 kPk 0 lim kPk 0 lim kPk 0 a Z b g ti 1ti i 1 ti 1ti f i 1 Z b f t dt lim kPk 0 n X g ti 1ti i 1 g t dt a Z b f t g t dt a Z b r t dt a Z b f t dt a g t dt a We simply map the operation of integration onto the components of the vector function Moreover we may use the FTC to evaluate the component integrals f x i 1x i Double Integrals i 1 Definition Let f x y be a continuous function over a rectangular region provided this limit of a Riemann sum exists When this occurs f is said to be integrable on a b R x y a x b c y d in the x y plane We define the double integral of f by slicing and dicing as it were Think of mincing that onion with your Ginsu knife That is split a b into m subintervals and c d into n subintervals The norm kPk of the resulting partition P is the length of the longest diagonal among the subrectangles of the partition We then form a double Riemann sum and take the limit as kPk shrinks to 0 If this limit exists we obtain the double integral of f over R Antiderivatives The scalar function F x is an antiderivative or indefinite integral of the scalar function f x on an interval I if and only if F 0 x f x for all x I A vector function R t is an antiderivative or indefinite integral of the vector function r t on an interval J if and only if we have R0 t r t for all t J Fundamental Theorem of Calculus FTC Part 2 Computing a definite integral as a limit of a Riemann sum is quite tedious But computing it with the FTC is easy Let f be a continuous Z b b function on a b Then f x d x F x F b F a a n X In other words Often the x i are equally spaced and we have a regular partition Let x i x i i x i be in the i th subinterval and 1x i x i x i 1 be the length of this subinterval The norm of P is defined by kPk max 1x i Now let the number of subintervals n increase indefinitely while the norm of P shrinks to 0 The definite integral of f from a to b is defined by n X lim ti 1ti f a P a x 0 x 1 x 2 x n 1 x n b f x d x lim i 1 n X kPk 0 f ti 1ti g ti 1ti i 1 n X kPk 0 b f ti g ti 1ti i 1 n X lim Z Definition Let f be a function defined on I a b Split a b into n subintervals whose endpoints constitute a partition b i 1 kPk 0 r ti 1ti n X Review of Single Integrals from Calc 1 Z n X lim ZZ R a f x y dA lim kPk 0 n m X X f x i y j 1x i 1y j i 1 j 1 where F is an antiderivative of f Vector Integals Iterated integrals Computing the exact value of a double integral by taking a limit of a double Riemann sum is very difficult even when it is possible Let r t f t g t a t b be a continuous vector function and P be a partition of a b The definite integral of r A practical way to evaluate multiple integrals is via interated integration where we compute single integrals in succession 1 Vector Double Integrals In other words we repeatedly compute antiderivatives and apply the Fundamental Theorem of Calculus working from inside out until we are finished Let r x y f x y g x y be a continuous function defined on a region D of Type 1 and or Type 2 The vector double integral of r over D is just what you d think ZZ ZZ r x y d A f x y g x y d A D D Z Z ZZ f x y d A g x y d A This mechanistic approach has been fully automated as in the smi stepwise multiple integration commands on the TI 89 and in MATLAB Accordingly this reduces the problem of computing multiple integrals to simply setting them up That said you ought to try some problems purely by hand to get some feeling for the work that you are being spared D D Just map the integration operation onto the components of the Fubini s Theorem If f is continuous on a rectangular region vector function Moreover we may use Fubini s Theorem on R x y a x b c y d then general regions to evaluate the component integrals ZZ Z bZ d Z dZ b f x y dA f x y d y d x f x y d x d y a R c c a Generalizations Nonrectangular regions Let s extend the aforementioned concepts A Type I region has the form If our vector function has three components then D x y a x b g1 x y g2 x Z b It is a region in the x y plane bounded on the left by the vertical line x a on the right by the vertical line x b below by the curve y g1 x and above by the curve y g2 x The double integral of the function f x y over Z b Z g2 x D may be computed as f x y d y d x a a a f t dt Z b Z b g t dt a a h t dt If f x y z is continuous on the rectangular box R x y z a x b c y d r z s then the triple integral of f over R is defined by Z sZ dZ b ZZZ f x y z dV f x y z d x d y dz g1 x A Type II region has the form D x y c y d h 1 y x h 2 y r R It is a region in the x y plane bounded on the below by the horizontal line y c above by …
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