(6.5) Length and area in polar coordinates

Transcription

1 86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx x x dx Centre of mss + ln 6 ( 9 ) m 8 ( + ln 6) m SH 6: 5 9 7: : (65) Length nd re in polr coordintes In this section we show how to clculte re nd rc lengths using polr coordintes Wheres in crtesin coordintes it ws nturl to use rectngles to define re in polr coordintes the nturl shpe to consider is wedge The re of the pictured wedge is r l π (re of circle of rdius r) r r So to find the re enclosed by the curve r f(θ) for θ θ θ we clculte θ r θ θ rf The digrm lso suggests ( l) ( r) + (r) ( ( r ) ) + r ( ) so tht rc length should be given by θ θ r + ( dr )

2 (65) Length nd re in polr coordintes 87 Note tht ll the rc length formulæ we hve seen cn be obtined by mnipulting the forml expression (dx) rc length + (dy) For exmple the connection between crtesin nd polr coordintes is given by x r cos θ y r sin θ so tht dx ( r sin θ + dr cos θ) dy (r cos θ + dr sin θ) Then (dx) + (dy) ( ( r sin θ + dr cos θ) + (r cos θ + dr sin θ)) / ( r sin t r sin θ cos θ dr ( dr ) + cos θ + r cos t + r sin θ cos θ dr ( dr ) + sin θ ) / ( dr ) r + Exmples () For the crdioid r + sin θ (α) Clculte the re enclosed (β) Find the length of the curve Solution (α) A π π π π + + r ( + sin θ) ( + sin θ + sin θ) ( cos θ) π (β) l ( + sin θ) + cos θ π

3 88 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS + sin θ π π π + sin θ (cos θ + sin θ ) [ NB the integrnd is positive on π π ] [ sin θ ] π 8 π * (b) Show tht the re enclosed by the smller loop of the limçon r + cos θ is π (Note tht we llow r to be negtive in this exmple) A π/ π/ π/ π/ ( + cos θ) ( + cos θ + cos θ) ( + cos θ + cos θ) [ θ + sin θ + sin θ ] π π/ π + π (c) Show tht the re inside the circle r 5 sin θ nd outside the limçon r + sin θ is 8π + Solution r 5 sin θ is x + (y 5 ) 5 Let θ be the ngle where they intersect Then + sin θ 5 sin θ So sin θ ; θ π 6 5π 6 / A (r r) / / (5 sin θ) ( + sin θ) (5 sin θ sin θ sin θ)

4 / / (66) Appendix for ATH : Centres of ss in Generl 89 (6 sin θ sin θ ) ( cos θ sin θ) 8π + (d) Find the length of the prbolic spirl r θ between θ nd θ π l r + ( dr ) θ + (θ) [ θ + θ θ θ + (θ + ) / ] π ( (π + ) / 8 ) 578 * (66) Appendix for ATH : Centres of ss in Generl The slicing ide cn be extended in the following wy to two-nd three-dimensionl objects with vrible density * (66) Plne Regions (or Lmins) with Vrible Density Suppose tht lmin corresponding to region R hs vrible density ρ(x y) mss/unit re If we slice the region t x the slice is line It will contribute (x) ρ(x y)dy to the totl mss Therefore the totl mss is given by b (x)dx x b The coordintes of the center of mss of the lmin re given by ( b ) xρ(x y)dy dx x ( b ) yρ(x y)dy dx y

5 9 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Compre these formule with the formul for the x-coordinte of the centre of mss of rod (see section 6) The expressions x ρ(x y)dy nd y ρ(x y)dy re clled the moments of the lmin bout the y xis nd the x xis respectively If the density ρ(x y) is constnt throughout the lmin then the center of mss is clled the centroid of the lmin * Exmple A lmin hs the shpe of the region in the first qudrnt tht is bounded by the grphs y sin x y cos x between x nd x π The density of the lmin is given by ρ(x y) y Find the coordintes of the centre of mss The mss of the strip is y ( ) π y sin x π y cos x x (x) ρ(x y) dy y dy cos x [ ] y cos x y dy sin x sin x ( cos x sin x ) so the totl mss is (x) dx x cos x dx The x coordinte of the center of mss is given by ( ) x y dy dx ( cos x sin x xy dy) dx / [x y ( cos x sin x ) dx [ ] π sin x ] y cos x y sin x dx x cos x dx π

6 Similrly y (66) Appendix for ATH : Centres of ss in Generl 9 ( π ) y y dy dx ( cos x [ y sin x ] y cos x ) y dy dx y sin x dx ( cos x sin x ) dx ( cos x( sin x) sin x( cos x) ) dx [ sin x sin x + cos x cos x ( Hence the center of mss hs the coordintes π ] π 8 9 ) 8 9 * (66) Three-Dimensionl Objects with Vrible Density Suppose tht closed nd bounded three-dimensionl object hs vrible density ρ(x y z) mss/unit volume The slice through x will be plne region nd we cn compute its contribution (x) to the totl mss by the sme method s we used to clculte the mss of lmin: the slice of through y will be line y which contributes m(x y) ρ(x y z)dz to (x) so (x) y m(x y)dy b x S xy S y x nd the totl mss of the object is b (x)dx The coordintes of the center of mss of the object re given by ( b ( ) ) x ρ(x y z)dz dy dx y x ( b ( ) ) y ρ(x y z)dz dy dx y y ( b ( ) ) z ρ(x y z)dz dy dx y z

7 9 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS where the multiple integrls on the numertor of x y nd z re moments of the object bout the coordinte plnes yz xz nd xy Note tht the centre of mss is independent of the choice of origin This mens tht clever choice of origin cn llow the symmetry of body to be exploited so s to reduce the mount of clcultion required to find the centre of mss