Question

Make a rough sketch of the region {(x, y) : 0 ≤ y ≤ x² + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} and find the area of the region, using the method of integration.

Answer 1

To find the point of intersections of the curve y = x² + 1 and the line y = x + 1,

we write x² + 1 = x + 1 `\implies` x(x – 1) = 0 `\implies` x = 0, 1.

So, the point of intersections P(0, 1) and Q(1, 2).

Area of the shaded region OPQRTSO = (Area of the region OSQPO + Area of the region STRQS)

= `int_0^1 (x^2 + 1)dx + int_1^2(x + 1)dx`

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

= `[(1/3 + 1) - 0] + [(2 + 2) - (1/2 + 1)]`

= `23/6`

Hence the required area is `23/6` sq units.