Find the area of the region enclosed by the parabola $x^{2}=y$, the line $y=x+2$ and $x-$ axis

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Find the area of the region enclosed by the parabola $x^{2}=y$ , the line $y=x+2$ and $x-$ axis

Find the area of the region enclosed by the parabola $x^{2}=y$ , the line $y=x+2$ and $x-$ axis

Solution:
Area of the region enclosed by the parabola, $x^{2}=y$ , the line, $y=x+2$ , and $x$ -axis is represented bv the shaded reaion $\mathrm{OABCO}$ as

Point of intersection of the parabola, $x^{2}=y$ , and the line, $y=x+2$ , is A $(-1,1)$
$\therefore$ Area $\mathrm{OABCO}=$ Area $(\mathrm{BCA})+$ Area COAC
$\begin{array}{l} =\int_{-2}^{-1}(x+2) d x+\int_{-1}^{0} x^{2} d x \\ =\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{-1}+\left[\frac{x^{3}}{3}\right]_{-1}^{0} \\ =\left[\frac{(-1)^{2}}{2}+2(-1)-\frac{(-2)^{2}}{2}-2(-2)\right]+\left[-\frac{(-1)^{3}}{3}\right] \\ =\left[\frac{1}{2}-2-2+4+\frac{1}{3}\right] \\ =\frac{5}{6} \text {sq. units } \end{array}$