evaluate the double integral (1+2y)dA, where D is the region bounded by the line

Question

evaluate the double integral (1+2y)dA, where D is the region bounded by the line y=x and the parabola y=x^2

Explanation / Answer

Points of intersection: 2x^2 = 1 + x^2 ==> x = -1, 1. For x in (-1, 1), we have 1 + x^2 > 2x^2. (This should be enough to sketch the region.) So, the integral equals ?(x = -1 to 1) ?(y = 2x^2 to 1 + x^2) (x + 2y) dy dx = ?(x = -1 to 1) (xy + y^2) {for y = 2x^2 to 1 + x^2} dx = ?(x = -1 to 1) [x(1 + x^2) + (1 + x^2)^2] - [x(2x^2) + (2x^2)^2] dx = 2 ?(x = 0 to 1) [(1 + x^2)^2 - (2x^2)^2] dx, even/odd arguments = 2 ?(x = 0 to 1) (1 + 2x^2 - 3x^4) dx = 2(x + 2x^3/3 - 3x^5/5) {for x = 0 to 1} = 32/15.

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