Question

Find the area of the region enclosed by the parabola \(x^2=y\),the line \(y=x+2\) and \(x-axis\)

Answer 1

The correct answer is:\(\frac{5}{6}units\)
The area of the region enclosed by the parabola,\(x^2=y\),the line,\(y=x+2\),and \(x-axis\) 
is represented by the shaded region OABCO as

The point of intersection of the parabola,\(x^2=y\),and the line,\(y=x+2\),is \(A(–1,1).\)
∴Area OABCO=Area(BCA)+Area COAC
\(=∫^{-1}_{-2}(x+2)dx+∫^0_{-1}x^2dx\)
\(=\bigg[\frac{x^2}{2}+2x\bigg]^{-1}_{-2}+\bigg[\frac{x^3}{3}\bigg]^0_{-1}\)
\(=[\frac{(-1)^2}{2}+2(-1)-(\frac{-2)^2}{2}-2(-2)]+[-\frac{(-1)^3}{3}]\)
\(=[\frac{1}{2}-2-2+4+\frac{1}{3}]\)
\(=\frac{5}{6}units\)