Aylanish sirtining yuzini hisoblash usullari

Аннотация

Ushbu maqolada aylanish sirtlarining yuzini hisoblash usullari keltirilgan va unga doir misollar ham berilgan. Kalit so‘zlar. Aylanish sirtlari, differensiallanuvchi funksiya, Ox o‘qi atrofida aylantirish natijasida hosil qilingan sirt, Gorizontal aylanma sirt yuzasini hisoblash, Vertikal aylanma sirt yuzasini hisoblash. Aytaylik, [a;b] kesmada manfiy bo‘lmagan uzluksiz differensiallanuvchi y=f(x) funksiya grafigini Ox o‘qi atrofida aylantirish natijasida sirt hosil qilingan bo‘lsin (1-rasm). Shu aylanish sirtining yuzini hisoblashda quyidagi formuladan foydalanamiz: b S  2 f (x) 1[f '(x)]2dx (1) a Bu aylanish sirtining yuzini topish formulasidir. i x x i a x x b i1 i 1-rasm Gorizontal aylanma sirt yuzasini hisoblash: > SurfaceOfRevolution(f(x), x=a..b, output=integral); 97 b 2 p | f ( x ) | 0d f ( x ) 12 C 1 dx a dx Vertikal aylanma sirt yuzasini hisoblash: > SurfaceOfRevolution(f(x), x=a..b, output=integral, axis=vertical); b 0d 12 2 p | x | f ( x ) C 1 dx a dx 1 misol. y  x funksiya grafigining x[1;2] kesmaga mos qismini Ox o‘qi atrofida aylantirishdan hosil bo‘lgan sirtning yuzini hisoblang. 1 Yechish. f(x)  x, f'(x)  . Bularni (1) ga qo‘yamiz. 2 x 2  1  2 2 1 S 2 x 1  dx 2 x dx  2 x 4 1 1 3  3 3 2 2   x 1   2  4     2 1     1 1     3  4  1 3  4   4   427 5 5 275 5      8,283. 3 8 8 6 >restart; with(plots): with(Student[Calculus1]): >f:=x->sqrt(x): >plot(f(x),x=1..2,color=red,style=line, thickness=2, title=`YOY`); 2 p x 1 C 4 x dx 1 x > SurfaceOfRevolution(f(x), x=1..2, output=integral); > SurfaceOfRevolution(f(x), x=1..2); 5 9 K 5 p C p 6 2 > evalf(%); 283154665 > SurfaceOfRevolution(f(x), x=1..2, output=plot); 98 2-misol. y  x2 funksiya grafigining x[1;2] kesmaga mos qismini: Oy o‘qi atrofida aylantirishdan hosil bo‘lgan sirtning yuzini hisoblang. >restart; with(plots): with(Student[Calculus1]): >f:=x->x^2: >plot(f(x),x=1..2,color=red,style=line, thickness=2, title=`YOY`); > SurfaceOfRevolution(f(x), x=1..2, output=integral, axis=vertical); 2 ∫ 2 p x 4 x 2 C 1 dx 1 > SurfaceOfRevolution(f(x),x=1..2,axis=vertical); 5 17 K 5 p C 17 p 6 6 > evalf(%); 30.84648972 > SurfaceOfRevolution(f(x), x=1..2, output=plot,axis=vertical); 3-misol. y  2 x2, y3  x2 chiziqlar bilan chegaralangan yassi figyrani Oy va Ox o’qiga atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblang. 1) Berilgan chiziqlar bilan chegaralangan yassi figyra: > restart; with(plots): with(Student[Calculus1]): > f2:=x->2-x^2: f1:=x->(x^2)^(1/3): > plot({f2(x),f1(x)},x=-2..2, y=0..2,color=[red,blue], style=line,thickness=2,title=`YUZA`); 2) figyrani Oy o’qiga atrofida aylanishsan hosil bo’lgan sirtni qurish: 99 > S1:=SurfaceOfRevolution(f2(x), x=0..1, output=plot, axis=vertical): S2:=SurfaceOfRevolution(f1(x), x=0..1, output=plot, axis=vertical): display({S2,S1}, axes=boxed, title=`SIRT`); 3) figyrani Oy o’qiga atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblash > SurfaceOfRevolution(f2(x), x=0..1, output=integral, axis=vertical)+SurfaceOfRevolution(f1(x), x=0..1, output= integral, axis=vertical); 1 ∫ 1 2 p x 4 x 2 C 1 dx C ó 2 p x 4 C 9 x ( 2 / 3 ) dx 0 õ 3 x ( 2 / 3 ) 0 > SurfaceOfRevolution(f2(x), x=0..1,axis=vertical)+ SurfaceOfRevolution(f1(x), x=0..1,axis=vertical); 1 28 8 8 p (5 5 K 1) C p 13 K p ln(2) C p ln(3 6 81 243 243 C 13) > evalf(%); 369550289 4) figurani Ox o’qiga atrofida aylanishsan hosil bo’lgan sirtni qurish. > S1:=SurfaceOfRevolution(f2(x), x=0..1, output=plot): S2:=SurfaceOfRevolution(f1(x), x=0..1, output=plot): display({S2,S1}, axes=boxed, title=`SIRT`); 5) figurani Ox o’qiga atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblash. > SurfaceOfRevolution(f2(x), x=0..1, output=integral)+ SurfaceOfRevolution(f1(x), x=0..1, output=integral); 100 1 ∫ K 2 p ( K 2 C x 2 ) 4 x 2 C 1 dx 0 1 C ∫ 2 p x ( 2 / 3 ) 4 C 9 x ( 2 / 3 ) dx 3 x ( 2 / 3 ) 0 > SurfaceOfRevolution(f2(x), x=0..1)+ SurfaceOfRevolution(f1(x), x=0..1); 33 23 33 01 1 1 128 p ln(2) C p 5 C p ln C 5 C p 16 16 32 2 4 1215 494 C 13 p 1215 > evalf(%); 171163206 3-misol. x2/3  y2/3  a2/3 astroida-chizig’i bilan chegaralangan egri chiziqli yassi yuzani Ox o’qi atrofida aylanishdan hosil bo’lgan sirt yuzasini hisoblash. Yechish: Ox o’qi atrofida aylanishdan hosil bo’lgan sirt yuzasini hisoblash: S 2y (x)2 (y)2dt x formulasiga asosida, astroidaning 101arametric tenglamasi x=acos3(t), y=asin3(t) (t )ga asosan S 22y (x)2 (y)2dt= x  /2 S 22y (x)2 (y)2dt 4asin3t 9a2cos4tsin2t9a2sin4t cos2t dt  x  0 /2 /2 12a2 /2 12a2 =4a  sin2t3asintcostdt 12a2  sin4td(sint) sin5t  5 0 5 0 0 1) astroida grafigini quramiz: > with(plots): > plot([1*cos(t)^3,1*sin(t)^3, t=0..2*Pi],thickness=2); 2) astroidani [0,π/2] kesmasiga mos qimining Ox o’qiatrofida aylanishdan hosil bo’lgan sirt grafigi. > restart; with(plots): with(Student[Calculus1]): > VolumeOfRevolution(sqrt(1^(2/3)-x^(2/3))^3, x=0..1, output=plot); 101 3) astroida-chizig’I bilan chegaralanga