Example 5 - Find area bounded by ellipse x2/a2 + y2/b2 = 1 - Area bounded by curve and horizontal or vertical line

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Slide18.JPG

  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Example 5 Find the area bounded by the ellipse 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 and the ordinates 𝑥=0 and 𝑥=𝑎𝑒, where, 𝑏2=𝑎2 (1 – 𝑒2) and e < 1 Required Area = Area of shaded region = Area BORQSP = 2 × Area OBPS = 2 × 0﷮𝑎𝑒﷮𝑦.𝑑𝑥﷯ We know that , 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯= 𝑎﷮2﷯ − 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑦﷮2﷯= 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ ∴ 𝑦=± ﷮ 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑎﷮2﷯− 𝑥﷮2﷯﷯﷯ As OBPS is in 1st quadrant 𝑦= 𝑏﷮𝑎﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ Required Area = 2 × 0﷮𝑎𝑒﷮𝑦.𝑑𝑥﷯ = 2 0﷮𝑎𝑒﷮ 𝑏﷮𝑎﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 2𝑏﷮𝑎﷯ 0﷮𝑎𝑒﷮ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 2𝑏﷮𝑎﷯ 1﷮2﷯𝑥 ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯+ 𝑎﷮2﷯﷮𝑎﷯ sin﷮−1﷯﷮ 𝑥﷮𝑎﷯﷯﷯﷮0﷮𝑎𝑒﷯ = 2𝑏﷮𝑎﷯ 𝑎𝑒﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑎𝑒﷯﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑎𝑒﷮𝑎﷯− 0﷮2﷯ ﷮ 𝑎﷮2﷯−0﷯− 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 0﷮𝑎﷯﷯﷯﷯﷯ = 2𝑏﷮𝑎﷯ 𝑎𝑒﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑎﷮2﷯ 𝑒﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑒﷯−0− 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 0﷯﷯﷯﷯ = 2𝑏﷮𝑎﷯ 𝑎𝑒﷮2﷯.𝑎 ﷮1− 𝑒﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮𝑒−0﷯﷯ = 2𝑏﷮𝑎﷯ 𝑎﷮2﷯𝑒﷮2﷯ ﷮1− 𝑒﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮𝑒﷯﷯ = 2𝑏﷮𝑎﷯ 𝑎﷮2﷯﷮2﷯﷯ 𝑒 ﷮1− 𝑒﷮2﷯﷯+ sin﷮−1﷯﷮𝑒﷯﷯ =𝑎𝑏 𝑒 ﷮1− 𝑒﷮2﷯﷯+ sin﷮−1﷯﷮𝑒﷯﷯ ∴ Required Area =𝒂𝒃 𝒆 ﷮𝟏− 𝒆﷮𝟐﷯﷯+ 𝒔𝒊𝒏﷮−𝟏﷯﷮𝒆﷯﷯ square units

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