Application of Integrals
Board: CBSE |Class:Class12
ComprehensivestudynotesforApplicationofIntegralsbyAjayYadav.
KeyConcepts
AreaUnderCurve
Areaboundedbyy=f(x),x-axis,x=a,x=b=∫an|f(x)|dx.Iff(x)≥0on[a,b],area=∫anf(x)dx.
AreaBetweenCurves
Areabetweeny=f(x)andy=g(x)wheref(x)≥g(x):A=∫an[f(x)-g(x)]dx.Findintersectionpointsforlimits.
Areaw.r.t.y-axis
Areaboundedbyx=f(y),y-axis,y=c,y=d:A=∫eg|f(y)|dy.
Important Formulas
| Area under curve | A = ∫anf(x)dx (f(x)≥0) |
| Between curves | A = ∫an[f(x)-g(x)]dx |
Solved Examples
Example 1: Find area under y=x² from x=0 to x=2.
Solution: A = ∫₀²x²dx = (x³/3)₀² = 8/3 sq units.
Example 2: Find area between y=x and y=x².
Solution: Intersection: x=x² ⇒ x=0,1. A = ∫₀¹(x-x²)dx = (x²/2-x³/3)₀¹ = 1/2-1/3 = 1/6 sq units.
Practice Questions
- Find area under y=sinx from 0 to π.
- Find area between y²=4x and x=4.
- Find area between y=x² and y=2x.
- Find area bounded by x²+y²=9 (circle).
- Find area of ellipse x²/16+y²/9=1.
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