Solving for Area Between Parabolas

Determine the bounded area between two parabolas by finding their intersection points and calculating the area enclosed. Follow a systematic approach independent of constants, consider translations to avoid negative areas, and evaluate the area accurately by tracking the curve shift. The mathematical process is detailed with images to aid understanding and visual representation.

• Parabolas
• Intersection Points
• Area Calculation
• Mathematics

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1. Area between Parabolas

2. ?(?) What s the area bounded by these parabolas? ?(?)

3. Let the functions be: Where do they intersect? ??2+ ?? + ? = ??2+ ?? + ? ? ? ?2= ? ? ?2= 1 ? = 1 ?? ? = 1 Notice that these values are independent of ?, ?and?. The area we require is given by: 1 ? ? = ??2+ ?? + ? ? ? = ??2+ ?? + ? ???? = ? ? ?(?) ? 1

4. ? ? + ? ?(?) We only need to evaluate: 1 ? ? + ? ?(?) ???? = ? ? ?(?) ? 1 despite part of its area being below the x-axis and therefore negative. Imagine both curves translated by the vector 0 4. The area wouldn t change and we would have no negative areas.

5. So what does ? ? ? ?look like? ?(?)-?(?) ?(?) ?(?)

6. So what does ? ? ? ?look like? ?(?)-?(?) ?(?) ?(?)

7. 1 ? ? ?2+ (? ?) ? ???? = 1 1 ? ? ?3 3 = + ? ? ? 1 ? ? 3 ? ? 3 = + ? ? ? ? =2 ? ? + 2? 2? 3 ???? =4 ? ? 3

8. B A D C

9. F E H G

10. J I L K

11. N M P O

13. Note to Teacher In all cases (? ?) is divisible by 3 to make it easier for you to check an answer.

14. Resources

15. Area between Parabolas What s the area bounded by these parabolas? ? ? = 2?2+ 3? + 4 ? ? = 4?2+ 3? 2 A Not accurately drawn SIC_24

16. Area between Parabolas What s the area bounded by these parabolas? ? ? = ?2 3? + 5 ? ? = 5?2 3? 1 B Not accurately drawn SIC_24

17. Area between Parabolas What s the area bounded by these parabolas? ? ? = 5?2 3? + 4 ? ? = 4?2 3? 5 C Not accurately drawn SIC_24

18. Area between Parabolas What s the area bounded by these parabolas? ? ? = 2?2+ 4? + 7 ? ? = 7?2+ 4? 2 D Not accurately drawn SIC_24

19. Area between Parabolas What s the area bounded by these parabolas? ? ? = ?2+ 4? + 5 ? ? = 5?2+ 4? 1 E Not accurately drawn SIC_24

20. Area between Parabolas What s the area bounded by these parabolas? ? ? = ?2+ ? + 7 ? ? = 7?2+ ? + 1 Not accurately drawn F SIC_24

21. Area between Parabolas What s the area bounded by these parabolas? ? ? = 2?2 5? + 8 ? ? = 8?2 5? + 2 Not accurately drawn G SIC_24

22. Area between Parabolas What s the area bounded by these parabolas? ? ? = 5?2 7? + 11 ? ? = 11?2 7? + 5 Not accurately drawn H SIC_24

23. Area between Parabolas What s the area bounded by these parabolas? ? ? = 2?2+ 3? + 4 ? ? = 4?2+ 3? 2 I Not accurately drawn SIC_24

24. Area between Parabolas What s the area bounded by these parabolas? ? ? = 3?2 3? + 9 ? ? = 9?2 3? 3 J Not accurately drawn SIC_24

25. Area between Parabolas What s the area bounded by these parabolas? ? ? = 3?2 2? + 9 ? ? = 9?2 2? 3 K Not accurately drawn SIC_24

26. Area between Parabolas What s the area bounded by these parabolas? ? ? = 3?2+ 4? + 3 ? ? = 3?2+ 4? 3 L Not accurately drawn SIC_24

27. Area between Parabolas What s the area bounded by these parabolas? ? ? = 5?2+ 7? 2 ? ? = 2?2+ 7? 5 M Not accurately drawn SIC_24

28. Area between Parabolas What s the area bounded by these parabolas? ? ? = 4?2 ? + 7 ? ? = 7?2 ? + 4 Not accurately drawn N SIC_24

29. Area between Parabolas What s the area bounded by these parabolas? ? ? = 5?2 5? + 8 ? ? = 8?2 5? + 5 Not accurately drawn O SIC_24

30. Area between Parabolas What s the area bounded by these parabolas? ? ? = ?2 7? + 23 ? ? = 23?2 7? 1 P Not accurately drawn SIC_24

31. What is the area bounded by these curves? What is the area bounded by these curves? ? ? = 2?2+ 3? + 4 ? ? = 4?2+ 3? 2 ? ? = ?2+ 4? + 5 ? ? = 5?2+ 4? 1 A E SIC_24 SIC_24 What is the area bounded by these curves? ? ? = ?2 3? + 5 ? ? = 5?2 3? 1 B What is the area bounded by these curves? ? ? = ?2+ ? + 7 ? ? = 7?2+ ? + 1 F SIC_24 SIC_24 What is the area bounded by these curves? ? ? = 5?2 3? + 4 ? ? = 4?2 3? 5 C What is the area bounded by these curves? ? ? = 2?2 5? + 8 ? ? = 8?2 5? + 2 G SIC_24 SIC_24 What is the area bounded by these curves? ? ? = 2?2+ 4? + 7 ? ? = 7?2+ 4? 2 D What is the area bounded by these curves? ? ? = 5?2 7? + 11 ? ? = 11?2 7? + 5 H SIC_24 SIC_24

32. What is the area bounded by these curves? What is the area bounded by these curves? ? ? = 2?2+ 3? + 4 ? ? = 4?2+ 3? 2 ? ? = 5?2+ 7? 2 ? ? = 2?2+ 7? 5 I M SIC_24 SIC_24 What is the area bounded by these curves? ? ? = 3?2 3? + 9 ? ? = 9?2 3? 3 J What is the area bounded by these curves? ? ? = 4?2 ? + 7 ? ? = 7?2 ? + 4 N SIC_24 SIC_24 What is the area bounded by these curves? ? ? = 3?2 2? + 9 ? ? = 9?2 2? 3 K What is the area bounded by these curves? ? ? = 5?2 5? + 8 ? ? = 8?2 5? + 5 O SIC_24 SIC_24 What is the area bounded by these curves? ? ? = 3?2+ 4? + 3 ? ? = 3?2+ 4? 3 L What is the area bounded by these curves? ? ? = ?2 7? + 23 ? ? = 23?2 7? 1 P SIC_24 SIC_24