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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.
Solution
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.
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