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Using integration, find the area of the region bounded by the line x - y + 2 = 0, the curve x = y and Y-axis.

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Given curves are

x - y + 2 = 0, ...........(i)

the curve x = y ..........(ii)

Consider x = y  or x2 = y, which represents the parabola whose vertex is (0, 0) and axis is Y-axis.

Now, the point of intersection of Eqs. (i) and (ii) is given by

But x = - 1 does not satisfy the Eq. (ii)

x = 2

Now, putting x = 2 in Eq. (ii), we get

2 = y or y = 4

Hence, the point of intersection is (2,4).

But we have actual equation of parabola x = y, it means a semi-parabola which is one right side of Y-axis.

The graph of given curve area shown below :

Clearly, area of bounded region = Area of region OABO

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