Question

Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x², 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.

Answer 1

The points of intersection of the parabola y = x² and the line y = x are (0, 0) and (1, 1).

Required Area = `int_0^1 y_(parabola) dx + int_1^2 y_"line"dx`

Required Area = `int_0^1 x^2 dx + int_1^2 x  dx`

= `[x^3/3]_0^1 + [x^2/2]_1^2`

= `1/3 + 3/2`

= `11/6`