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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.
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Solution
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`
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