Quadratic Segments

Calculate the areas of the segments Quadratic Segments 𝒚 𝑪 𝒙=−𝟕 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟓 𝑫 𝑨 𝟒,𝟐𝟒 𝟏 𝒙 𝟐 (Not to scale)

Quadratic Segments 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 Let’s look at the general case. 𝑥=𝑝+𝐿

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

Quadratic Segments Area of Trapezium 1 2 × 𝑎 𝑝 2 +𝑏𝑝+𝑐+𝑎 𝑝+𝐿 2 +𝑏 𝑝+𝐿 +𝑐 ×𝐿 1 2 𝑎 2𝑝 2 +2𝑝𝐿+ 𝐿 2 +𝑏 2𝑝+𝐿 +2𝑐 𝐿 𝑎 2 𝑝 2 𝐿+2𝑝 𝐿 2 + 𝐿 3 2 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿 Area under Curve 𝑝 𝑝+𝐿 𝑎 𝑥 2 +𝑏𝑝+𝑐 𝑑𝑥= 𝑎 𝑥 3 3 + 𝑏 𝑥 2 2 +𝑐𝑥 𝑝+𝐿 𝑝 = 𝑎 𝑝+𝐿 3 3 + 𝑏 𝑝+𝐿 2 2 +𝑐 𝑝+𝐿 − 𝑎 𝑝 3 3 + 𝑏 𝑝 2 2 +𝑐𝑝 = 𝑎 3 𝑝 2 𝐿+3𝑝 𝐿 2 + 𝐿 3 3 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿

Quadratic Segments Area of Trapezium 𝑎 2 𝑝 2 𝐿+2𝑝 𝐿 2 + 𝐿 3 2 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿 Area under Curve 𝑎 3 𝑝 2 𝐿+3𝑝 𝐿 2 + 𝐿 3 3 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿 Area of Segment 𝑎 2 𝑝 2 𝐿+2𝑝 𝐿 2 + 𝐿 3 2 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿 − 𝑎 3 𝑝 2 𝐿+3𝑝 𝐿 2 + 𝐿 3 3 + 𝑏 2𝑝𝐿+ 𝐿 2 2 +𝑐𝐿 𝑎 𝐿 3 2 − 𝑎 𝐿 3 3 𝑎 𝐿 3 6

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

RESOURCES

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟕 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟓 𝑫 𝑨 𝟒,𝟐𝟒 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟓 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟑 𝑫 𝑨 𝟒,𝟐𝟒 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟑 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟏 𝑫 𝑨 𝟒,𝟐𝟒 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟑 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟕 𝟐 𝒙=−𝟏 𝑫 𝑨 𝟓,𝟒𝟓 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟓 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟕 𝟐 𝒙=−𝟑 𝑫 𝑨 𝟓,𝟒𝟓 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟔 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟕 𝟐 𝒙=−𝟒 𝑫 𝑨 𝟓,𝟒𝟓 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟔 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟒 𝑫 𝑨 𝟑,𝟏𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟒 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟐 𝑫 𝑨 𝟑,𝟏𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟑 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟏 𝑫 𝑨 𝟑,𝟏𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟑 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟏 𝑫 𝑨 𝟒,𝟒𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟓 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟑 𝑫 𝑨 𝟒,𝟒𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟒 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟐 𝑫 𝑨 𝟒,𝟒𝟖 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟒 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟐 𝑫 𝑨 𝟑,𝟐𝟕 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟔 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟒 𝑫 𝑨 𝟑,𝟐𝟕 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟕 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟓 𝟐 𝒙=−𝟓 𝑫 𝑨 𝟑,𝟐𝟕 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟕 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟓 𝑫 𝑨 𝟒,𝟕𝟐 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟓 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟑 𝑫 𝑨 𝟒,𝟕𝟐 𝟏 𝒙 𝟐 (Not to scale)

Calculate the areas of the segments SIC_40 Quadratic Segments 𝒚 𝑪 𝒙=−𝟑 parabola 𝟑 Calculate the areas of the segments 1, 2 and 3 𝑩 𝒙=𝟔 𝟐 𝒙=−𝟏 𝑫 𝑨 𝟒,𝟕𝟐 𝟏 𝒙 𝟐 (Not to scale)