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प्रश्न
Find the area of the region bounded by the parabola: y = 4 – x2 and the X-axis.
उत्तर
The equation of the parabola is y = 4 – x2
∴ x2 = 4 – y, i.e. (x – 0)2 = – (y – 4)
It has vertex at P(0, 4).
For points of intersection of the parabola with X-axis, we put y = 0 in its equation.
∴ 0 = 4 – x2
∴ x2 = 4
∴ x = ± 2.
∴ The parabola intersect the X-axis at A(– 2, 0) and B(2, 0)
Required area = Area of the region APBOA
= 2[Area of the region OPBO]
= `2int_0^2 y dx, "where" y = 4 - x^2`
= `2int_0^2 (4 - x^2)dx`
= `8int_0^2 1dx - 2 int_0^2x^2 dx`
= `8[x]_0^2 - 2[x^3/3]_0^2`
= `8(2 - 0) - 2/3(8 - 0)`
= `16 - 16/3`
= `32/3` sq units.
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