Question

Find the area of the region bounded by the parabola: y = 4 – x² and the X-axis.

Answer 1

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.