Understanding Quadratic Segments and Their Areas

Discover the relationship between parabolas and segments in the realm of quadratic geometry. Learn how to calculate areas of different segments based on the trapezium method and the area under the curve, while exploring why the area only depends on specific variables. Dive into the essence of quadratic segments through insightful explanations and visual aids.

• Quadratic geometry
• Parabolas
• Area calculation
• Segment analysis
• Mathematical concepts

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1. Quadratic Segments

2. Quadratic Segments ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

3. Quadratic Segments ? = ??2+ ?? + ? Let s look at the general case. ? = ? + ? ? ? = ?

4. Quadratic Segments The area of the segment is given by: area of the trapezium of width ? created by the chord and the ? axis minus area under the curve.

5. Quadratic Segments Area of Trapezium 1 2 ??2+ ?? + ? + ? ? + ?2+ ? ? + ? + ? ? 1 2? 2?2+ 2?? + ?2+ ? 2? + ? + 2? ? ? 2?2?+2??2+?3 +? 2??+?2 + ?? 2 2 Area under Curve 2+ ??? + ? ??3 3+??2 ?+???2+ ?? + ? ?? = ? ? ? ?+?3 3 +? ?+?2 ??3 3+??2 = + ? ? + ? 2+ ?? 2 ? 3?2?+3??2+?3 +? 2??+?2 = + ?? 3 2

6. Quadratic Segments Area of Trapezium ? 2?2? + 2??2+ ?3 2 +? 2?? + ?2 + ?? 2 Area under Curve ? 3?2? + 3??2+ ?3 3 Area of Segment +? 2?? + ?2 + ?? 2 ? 2?2? + 2??2+ ?3 2 +? 2?? + ?2 ? 3?2? + 3??2+ ?3 3 +? 2?? + ?2 + ?? + ?? 2 2 ??3 2 ??3 3 ??3 6

7. Quadratic Segments Can you explain why the area only depends on ? and ?? Clearly the constant ? is irrelevant since a vertical translation in the ? direction is not going to change the area. Less obvious is the irrelevance of the constant ?. Adding the term ?? applies a vertical shear to the curve ? = ??2, which does not change its area (see Parabola in Parallelogram, SIC_30). So only ? and ? are relevant. The wider the segment (in the ? direction) the greater the area, obviously and the greater the stretch in the ? direction the greater the area.

9. RESOURCES

10. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

11. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

12. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

13. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

14. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

15. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

16. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

17. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

18. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

19. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

20. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

21. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

22. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

23. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

24. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

25. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

26. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)

27. Quadratic Segments SIC_40 ? ? ? = ? parabola Calculate the areas of the segments 1, 2 and 3 ? ? ? = ? ? ? = ? ? ? ?,?? ? ? ? (Not to scale)