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प्रश्न
Find the area of the ellipse `x^2/1 + y^2/4` = 1, in first quadrant
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उत्तर
Given equation of the ellipse is `x^2/1 + y^2/4` = 1.
∴ `y^2/4` = 1 − x2
∴ y2 = 4(1 – x2)
∴ y = `+- 2sqrt(1 - x^2)`
∴ y = `2sqrt(1 - x^2)` ......[∵ In first quadrant, y > 0]
∴ Required area
= `int_0^1 y "d"x`
= `int_0^1 2sqrt(1 - x^2) "d"x`
= `2[x/2 sqrt(1 - x^2) + 1^2/2 sin^-1(x/1)]_0^1`
= `2[0 + 1/2 sin^-1 (1) - 0]`
= `2[1/2 (pi/2)]`
= `pi/2` sq.units
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