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Question
Find the area of the region bounded by the parabola y2 = 2x and the straight line x – y = 4.
Solution
The intersecting points of the given curves are obtained by solving the equations x – y = 4 and y2 = 2x for x and y.
We have y2 = 8 + 2y
i.e., (y – 4)(y + 2) = 0
Which gives y = 4, –2 and x = 8, 2.
Thus, the points of intersection are (8, 4), (2, –2).
Hence Area = `int_(-2)^4 (4 + y - 1/2 y^2)"d"y`
= `|4y + y^2/2 - 1/6 y^3|_-2^4`
= 18 sq.units.
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