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Ex 8.1, 5 - Find area by ellipse x2/4 + y2/9 =1 - Class 12 - Ex 8.1

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Ex 8.1, 5 Find the area of the region bounded by the ellipse 𝑥﷮2﷯﷮4﷯+ 𝑦﷮2﷯﷮9﷯=1 Equation of given Ellipse is :- 𝑥﷮2﷯﷮4﷯+ 𝑦﷮2﷯﷮9﷯=1 𝑥﷮2﷯﷮ 2﷯﷮2﷯﷯+ 𝑦﷮2﷯﷮ 3﷯﷮2﷯﷯=1 Area of Ellipse = Area of ABCD = 2 × [Area of BCD] = 2 × −2﷮2﷮𝑦 .𝑑𝑥﷯ We know that 𝑥﷮2﷯﷮4﷯+ 𝑦﷮2﷯﷮9﷯=1 𝑦﷮2﷯﷮9﷯=1− 𝑥﷮2﷯﷮4﷯ 𝑦﷮2﷯﷮9﷯= 4 − 𝑥﷮2﷯﷮4﷯ 𝑦﷮2﷯= 9﷮4﷯ 4− 𝑥﷮2﷯﷯ Taking Square Root on Both Sides 𝑦=± ﷮ 9﷮4﷯ 4−4 𝑥﷮2﷯﷯﷯ 𝑦 =± 3﷮2﷯ ﷮4− 𝑥﷮2﷯﷯ Since , BCD is above x-axis ∴ 𝑦= 3﷮2﷯ ﷮4− 𝑥﷮2﷯﷯ Area of Ellipse = 2 × −2﷮2﷮𝑦 .𝑑𝑥﷯ = 2 × −2﷮2﷮ 3﷮2﷯ ﷮4− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 3 −2﷮2﷮ ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 3 𝑥﷮2﷯ ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯+ 2﷯﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑥﷮2﷯﷯﷯﷮−2﷮2﷯ = 3 2﷮2﷯ ﷮ 2﷯﷮2﷯− 2﷯﷮2﷯﷯− −4﷯﷮2﷯ ﷮ 2﷯﷮2﷯− −2﷯﷮2﷯﷯+2 sin﷮−1﷯﷮ 2﷮2﷯﷯−﷯ sin﷮−1﷯ −2﷮2﷯﷯﷯ = 3 1.0+1.0+2 sin﷮−1﷯﷮ 1﷯−2 𝒔𝒊𝒏﷮−𝟏﷯﷮ −𝟏﷯﷯﷯﷯ = 3 0+2 sin﷮−1﷯﷮ 1﷯−2(− 𝒔𝒊𝒏﷮−𝟏﷯﷮ 𝟏﷯﷯)﷯﷯ = 3 0+2 sin﷮−1﷯﷮ 1﷯+2 sin﷮−1﷯﷮ 1﷯﷯﷯﷯ = 3 4 𝒔𝒊𝒏﷮−𝟏﷯﷮ 𝟏﷯﷯﷯ = 3 × 4 × 𝝅﷮𝟐﷯ = 6π ∴ Area of Ellipse = 6π square units

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