๐ Key Concepts
Area Under Curve- Bounded by curve y = f(x)
- Bounded by x-axis
- Between given interval [a,b]
- Area = โซโแตf(x)dx if f(x) โฅ 0
Area Between Two Curves- Area between y = f(x) and y = g(x)
- Over interval [a,b]
- Intersection points important
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๐ฏ Key Formulas
Area formulas:
- Between curve & x-axis: โซโแต|f(x)|dx
- Between two curves: โซโแต[f(x) - g(x)]dx
- If curves intersect: Find points first
- For negative functions: Use absolute value
- Region bounded by x = g(y): โซ๐แต|gโปยน(y)|dy
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โ ๏ธ Common Mistakes to Avoid
Not considering negative areasWrong interval of integrationForgetting to find intersection pointsNot identifying upper and lower curvesUsing wrong formula for vertical curvesNot using absolute value when neededโ
๐ Knowledge Prerequisites
Integration techniquesFinding zeros of functionsGraphing skillsUnderstanding function behaviorSolving equationsโ
๐ก Tips for Students
Always sketch the regionFind intersection points firstCheck if function is positive/negativeBreak complex regions into simpler partsVerify results with approximate calculationsConsider symmetry if presentโ
๐ Practice Recommedations
Simple area calculationsAreas with negative functionsRegions between multiple curvesMixed regions (horizontal/vertical)Word problems on areasApplications in physics/economicsโ
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