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  • Find the area of the region enclosed by the parabola x^2 = y, the line y = x + 2 and the x-axis.

Q : 10        Find the area of the region enclosed by the parabola \small x^2=y, the line \small y=x+2  and the \small x-axis.
 

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best_answer

We have to find the area of the shaded region BAOB

O is(0,0)

The line and the parabola intersect in the second quadrant at (-1,1)

The line y=x+2 intersects the x axis at (-2,0)

\\ar(BAOB)=ar(BAC)+ar(ACO)\\ =\int_{-2}^{-1}(x+2)dx+\int_{-1}^{0}(x^{2})dx\\ =[\frac{x^{2}}{2}+2x]_{-2}^{-1}+[\frac{x^{3}}{3}]_{-1}^{0}\\ =(\frac{1}{2}-2)-(2-4)+0-(-\frac{1}{3})\\ =\frac{5}{6}\ units

The area of the region enclosed by the parabola \small x^2=y, the line \small y=x+2  and the \small x-axis is 5/6 units.

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Sayak

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