9-bob. Aniq integral

4. Aniq integralning geometrik masalalarga tatbiqlari

1. 
   y=f(x) funksiya grafigi, x=a va x=b vertikal chiziqlar hamda y=0 gorizontal to‘gri chiziq bilan chegaralangan egri chiziqli trapetsiyaning S yuzini umumiy holda hisoblash formulasi qayerda to‘g‘ri ko‘rsatilgan ?
   

A) . B) . C) .
D) . E) .

2. 
   y=x³ , x=0, x=2 va y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping .
   

A) S=1. B) S=2. C) S=3. D) S=4. E) S=5.

3. 
   y=x³ , x=–2, x=2 va y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping .
   

A) S=0. B) S=2. C) S=4. D) S=6. E) S=8.

4. 
   y=sinx , x=0, x=π/2 va y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping .
   

A) S=1. B) S=2. C) S=3. D) S=4. E) S=5.

5. 
   y=sinx , x=0, x=π va y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping .
   

A) S=1. B) S=2. C) S=3. D) S=4. E) S=5.

6. 
   y=cosx , x=0, x=π va y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping .
   

A) S=0. B) S=1. C) S=2. D) S=3. E) S=4.

7. 
   Quyidagi y=f(x) funksiya grafigi, x=0 va x=2 hamda y=0 chiziqlar bilan chegaralangan egri chiziqli trapetsiyaning S yuzini toping:
   

.
A) S=2. B) S=8/3. C) S=17/6. D) S=20/7. E) S=5.

8. 
   y=f(x) va y=g(x), f(x)≤g(x) , funksiyalarning grafiklari, x=a va x=b vertikal chiziqlar bilan chegaralangan geometrik shaklning S yuzini hisoblash formulasi qayerda to‘g‘ri ko‘rsatilgan ?
   

A) . B) . C) .
D) . E) .

9. 
   y=x² va y=x⁴ egri ciziqlar bilan chegaralangan shaklning S yuzini toping.
   

A) S=4. B) S=2. C) S=3/10. D) S=2/15. E) S=3/8.

10. 
    y=f(x) va y=g(x), f(x)≤g(x) , funksiyalarning grafiklari, x=a va x=b vertikal chiziqlar bilan chegaralangan geometrik shaklning S yuzini hisoblash formulasi qayerda noto‘g‘ri ko‘rsatilgan ?
    

A) . B) .
C) . D) .
E) barcha formulalar S yuzani to‘g‘ri ifodalaydi .

11. 
    Qutb koordinatalat sistemasida r=r(φ) tenglama berilgan L egri chiziq va φ=α , φ=β (α<β) radius vektorlar bilan chegaralangan sektor yuzi S aniq integral orqali hisoblanadigan formula qayerda to’gri ko’rsatilgan?
    

A) . B) . C) .
D) . E) .

12. 
    Qutb koordinatalar sistemasida r=3φ tenglama berilgan Arximed spirali va φ=0 , φ=2π radius vektorlar bilan chegaralangan sektorning S yuzini toping.
    

A) S=8π³ . B) S=10 π³ . C) S=12π³ . D) S=14 π³ . E) S=16 π³ .

13. 
    Qutb koordinatalar sistemasida tenglama berilgan va lemniskata deb ataluvchi egri chiziq hamda φ=0 , φ=2π radius vektorlar bilan chegaralangan sektorning S yuzini toping.
    

A) S=4π³ . B) S=4π² . C) S=4π . D) S=4 . E) S=4 .

14. 
    y=f(x), x [a, b], funksiya grafigidan iborat yoyning l uzunligini hisoblash formulasini ko‘rsating .
    

A) . B) . C) .
D) . E) .

15. 
    x=φ(t) , y=ψ(t), α≤t≤β, parametrik tenglamasi bilan berilgan egri chiziq yoyi l uzunligini hisoblash formulasini ko‘rsting .
    

A) . B) . C) .
D) . E) .

16. 
    Parametrik tenglamasi x=a(t–sint), y=a(1–cost) (0≤t≤2π) bo’lgan egri chiziq yoyining uzunligini toping.
    

A) 2a. B) 4a. C) 6a. D) 8a. E) 10a.