Integration in polar coordinates

Transcription

1 Pola Coodinates Integation in pola coodinates Pola coodinates ae a diffeent wa of descibing points in the plane. The pola coodinates (, θ) ae elated to the usual ectangula coodinates (, ) b b = cos θ, = sin θ The figue below shows the standad pola tiangle elating,, and θ. θ Because cos and sin ae peiodic, diffeent (, θ) can epesent the same point in the plane. The table below shows this fo a few points. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, θ) (, ) (, π/) (, ) (, π/4) (, 3π/4) (, 5π/4) (, π/) (, θ) (, π) (, 9π/4) (, π/4) (, 7.) (, θ) (, 4π) In fact, ou can add an multiple of π to θ and the pola coodinates will still epesent the same point. Because θ is not uniquel specified it s a little tickie going fom ectangula to pola coodinates. The equations ae easil deduced fom the standad pola tiangle. = +, θ = tan (/). We use quotes aound tan to indicate it is not a single valued function. The aea element in pola coodinates In pola coodinates the aea element is given b da = d dθ. The geometic justification fo this is shown in b the following figue. Δθ ΔA Δθ Δ The small cuv ectangle has sides Δ and Δθ, thus its aea satisfies ΔA (Δ)( Δθ). As usual, in the limit this becomes da = d dθ.

2 Double integals in pola coodinates The aea element is one piece of a double integal, the othe piece is the limits of integation which descibe the egion being integated ove. Finding pocedue fo finding the limits in pola coodinates is the same as fo ectangula coodinates. Suppose we want to evaluate d dθ ove the egion shown. + = = /(cos θ + sin θ) = (The integand, including the that usuall goes with d dθ, is ielevant hee, and theefoe omitted.) As usual, we integate fist with espect to. Theefoe, we. Hold θ fied, and let incease (since we ae integating with espect to ). As the point moves, it taces out a a going out fom the oigin.. Integate fom the -value whee the a entes to the -value whee it leaves. This gives the limits on. 3. Integate fom the lowest value of θ fo which the coesponding a intesects to the highest value of θ. To follow this pocedue, we need the equation of the line in pola coodinates. We have + = cos θ + sin θ =, o = cos θ + sin θ. This is the value whee the a entes the egion; it leaves whee =. The as which intesect lie between θ = and θ = π/. Thus the double iteated integal in pola coodinates has the limits π/ /(cos θ+sin θ) d dθ. Eample: Find the mass of the egion shown if it has densit δ(, ) = (in units of mass/unit aea) In pola coodinates: δ = cos θ sin θ. Limits of integation: (adial lines sweep out ): inne (fi θ): < <, oute: < θ < π/3. π/3 Mass M = δ(, ) da = cos θ sin θ dθ d θ= = 4 3 Inne: cos θ sin θ d = cos θ sin θ = 4 cos θ sin θ 4 π/3 π/3 3 Oute: M = 4 cos θ sin θ dθ = sin θ =. π/3

3 Eample: Let I= ( d d. Compute I using pola coodinates. + ) 3/ Answe: Hee ae the steps we take. Daw the egion. Find limits in pola coodinates: Inne (fi θ): sec θ < < sec θ, oute: < θ < π/4. π/4 sec θ I = d dθ. θ= =sec θ 3 Compute the integal: sec θ Inne: d = sec θ = cos θ. sec θ sec θ π/4 π/4 Oute: I = cos θ dθ = sin θ =. 4 = π/4 = o = sec θ Eample: Find the volume of the egion above the -plane and below the gaph of z =. You should daw a pictue of this. In pola coodinates we have z = above the inside of the unit disk. π volume V = ( ) d dθ. Inne integal: ( ) d = 4 = 4. π π Oute integal: V = dθ =. 4 and we want the volume unde the gaph and Galle of pola gaphs ( = f(θ)) A point P is on the gaph if an epesentation of P satisfies the equation. Eamples: π/3 Cicle centeed on : Vetical line = Hoizontal line = a: θ = π/3 = = sec θ. = / sin θ.

4 Eample: Show the gaph of = a cos θ is a cicle of adius a centeed at (a, ). Some simple algeba gives = a cos θ = a + = a ( a) + = a. This is a cicle o adius a centeed at (a, ). Note: we can detemine fom the gaph that the ange of theta is π/ θ π/. a a = a cos θ π/ θ π/. = a sin θ θ π. Waning: We can use negative values of fo plotting. You should neve use it in integation. In integation it is bette to make use of smmet and onl integate ove egions whee is positive. Hee ae a few moe cuves. Cadiod: = a( + cos θ) Limaçon: = a( + b cos θ) (b > ) Lemniscate: = a cos θ Fou leaved ose: = a sin θ

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