# HARVARD MATH 21A - 2d Integrals (2 pages)

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## 2d Integrals

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## 2d Integrals

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Harvard University
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Math 21a - Multivariable Calculus
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Lecture 17 3 19 2004 2D INTEGRALS Math21a O Knill HOMEWORK Section 12 1 16 26 30 44 Section 12 4 44 Rb 1D INTEGRATION IN 100 WORDS If f x is a continuous function then a f x dx can be defined as a P limit of the Riemann sum fn x n1 xk a b f xk for n with xk k n This integral divided by b a is the average of f on a b The integral can be interpreted as an signed Rarea under the graph x of f If f x 1 the integral is the length of the interval The function F x a f y dy is called an anti derivative of f The fundamental theorem of calculus states F 0 x f x Unlike the derivative Rx 2 anti derivatives can not always be expressed in terms of known functions An example is F x 0 e t dt Often the anti derivative can be found Example f x cos 2 x cos 2x 1 2 F x x 2 sin 2x 4 F x F x dx F x AVERAGES MEAN www worldclimate com gives the following data for the average monthly rainfall in mm for Cambridge MA USA 42 38 North 71 11 West 18m Height Jan 93 9 Feb 88 6 Mar 83 3 Apr 67 0 May 42 9 Jun 26 4 Jul 27 9 Aug 83 8 Sep 35 5 Oct 61 4 The average 860 3 12 71 7 is a Rieman sum integral Nov 166 8 Dec 82 8 150 125 100 75 50 25 1 2 3 4 5 6 7 8 9 10 11 12 2D R INTEGRATION If f x y is a continuous Pfunction of two variables on a region R the integral f x y dxdy can be defined as the limit n12 i j xi j R f xi yj with xi j i n j n when n goes to R infinity If f x y 1 then the integral is the area of the region R The integral divided by the area of R is the average value of f on R For many regions the integral can be calculated as a double R b R d x integral a c x f x y dy dx In general the region must be split into pieces then integrated seperately RR One can interpret f x y dydx as the volume of solid below the graph of f and above R in the x y plane R As in 1D integration the volume of the solid below the x y plane is counted negatively EXAMPLE Calculate Z 1 0 Z RR 2 0 R f x y dxdy where f x y 4x2 y 3 and where R is the rectangle 0 1 0 2 4x2 y 3 dy dx Z 1 0 x2 y 4 20 dx Z 1 0 x2 16 0 dx 16x3 3 10 16 3 RbRd FUBINI S THEOREM a c f x y dxdy TYPES OF REGIONS RR R b R g x f dA a g12 x f x y dydx R type I region R b R h y RR f dA a h12 y f x y dxdy R type II region RdRb c a f y x dydx Let R be the triangle 1 x 0 1 y 0 y x Calculate REXAMPLE R x2 e dxdy R R1 R1 2 2 ATTEMPT 0 y e x dx dy We can not solve the inner integral because e x has no anti derivative in terms of elementary functions R1 Rx R1 1 x2 2 2 IDEA Switch order 0 0 e x dy dx 0 xe x dx e 2 10 1 e2 0 316 A special case of switching the order of integration is Fubini s theorem If you can t solve a double integral try to change the order of integration QUANTUM MECHANICS In quantum mechanics the motion of a particle like an electron in the plane is determined by a function u x y the wave function Unlike in classical mechanics theR position of a particle 2 is R given in2 a probabilistic way only If R is a region and u is normalized so that u dxdy 1 then u x y dxdy is the probability that the particle is in R R EXAMPLE Unlike a classical particle a quantum particle in a box 0 0 can have a discrete set of energies only This is the reason for the name quantum If u xx uyy u then a particle of mass m has the energy E h 2 2m A function u x y sin kx sin ny represents a particle of energy k 2 n2 h 2 2m Let us assume k 2 and n 3 from now on Our aim is to find the probability that the particle with energy 13h 2 2m is in the middle 9 th R 3 2 3 3 2 3 of the box SOLUTION We first have to normalize u2 x y sin2 2x sin2 3y so that the average over the whole square is 1 Z Z sin2 2x sin2 3y dxdy A 0 0 this integral we Rfirst determine the inner integral RTo calculate 2 2 sin 2x sin 3y dx sin2 3y 0 sin2 2x dx 2 sin2 3y the fac0 R tor sin2 3y is treated as a constant Now A 0 2 sin2 3y dy 2 4 so that the probability amplitude function is f x y 2 2 4 2 sin 2x sin 3y 0 3 3 0 2 0 1 0 0 2 1 1 2 3 0 The probability that the particle is in R is slightly smaller than 1 9 1 A Z f x y dxdy R Z 2 3 Z 2 3 4 sin2 2x sin2 3y dxdy 2 3 3 4 2 3 2 3 4x sin 4x 8 3 6x sin 6x 12 3 2 1 9 1 4 3 The probability is slightly smaller than 1 9 WHERE DO DOUBLE INTEGRALS OCCUR compute areas compute averages Examples average rain fall or average population in some area probabilities Expectation of random variables RR quantum mechanics Rprobability of particle being x2 y 2 x y dxdy R R R R in a region find moment of inertia RR find center of mass R x x y dxdy M R y x y dxdy M with M dxdy R compute some 1D integrals INTEGRALS are defined similarly and covered later in detail RTRIPLE RR 1 dxdydz is a volume R Fubinis theorem generalizes

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