The area (in sq. units) bounded by $x^2=y, y=x+2$ and the $\mathrm{X}$-axis is

Question

Aicharya · Accepted Answer

Indicate the region bounded by the curves $x^2=y, y=x+2$ and $X$-axis and obtain the area enclosed by them. The parabola and line meet in points $P(-1,1)$ and $Q(2,4)$. The line cuts the axes in $(-2,0)$ and $B(0,2)$. Here, we have to find the area bounded by the curves and $X$-axis i.e. area $O R A P Q$ i.e. area $A P R+$ area $P R O$

$\begin{aligned} & =\int_{-2}^{-1} y_{\text {line }} d x+\int_{-1}^0 y_{\text {parabola }} d x \\ & =\int_{-2}^{-1}(x+2) d x+\int_{-1}^0 x^2 d x \\ & =\left[\frac{x^2}{2}+2 x\right]_{-2}^{-1}+\left[\frac{1}{3} x^3\right]_{-1}^0 \\ & =\frac{1}{2}+\frac{1}{3}=\frac{5}{6} \text { sq units } \end{aligned}$

The area (in sq. units) bounded by \(x^2=y, y=x+2\) and the \(\mathrm{X}\)-axis is

Solution