Question

The area bounded by the curve $y=x^2+4x+5$, the axes of co-ordinates and the minimum ordinate is

Answer 1

Given $y=x^2+4x+5$

differentiating the equation

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=2x+4 \\ \because \frac{dy}{dx}=0 \\ \Rightarrow 2x+4=0 \\ \Rightarrow x=-2 \end{gathered}$$

Required area $={\int }_{-2}^{0}ydx$

Required area $=$${\int }_{-2}^0{(x}^2+4x+5)dx$

$$\begin{gathered} ={\left[\frac{x^3}{3}+2x^2+5x\right]}_{-2}^0 \\ =\left[0+0+0\right]-\left[\frac{-8}{3}+8-10\right] \\ =0-\left[\frac{-8+24-30}{3}\right] \\ =-\left[\frac{-14}{3}\right] \\ =\frac{14}{3}sq.units \end{gathered}$$

Hence, the area is $\frac{14}{3}sq.units$