Question

The area of the bounded by the curve y = x² + x, x-axis and lines x = 2 and x = 5 is equal to __________________.

Answer 1

To find: area of the region bounded by the curve y = x² + x, x-axis and lines x = 2 and x = 5

The required area of the region = ${\int }_2^{5}ydx$
Thus,
$$\begin{gathered} \mathrm{Required}\mathrm{area}={\int }_2^{5}ydx \\ ={\int }_2^5\left(x^2+x\right)dx \\ ={\left(\frac{x^3}{3}+\frac{x^2}{2}\right)}_2^5 \\ =\left(\frac{5^3}{3}+\frac{5^2}{2}\right)-\left(\frac{2^3}{3}+\frac{2^2}{2}\right) \\ =\left(\frac{125}{3}+\frac{25}{2}\right)-\left(\frac{8}{3}+\frac{4}{2}\right) \\ =\left(\frac{125-8}{3}+\frac{25-4}{2}\right) \\ =\frac{117}{3}+\frac{21}{2} \\ =39+\frac{21}{2} \\ =\frac{78+21}{2} \\ =\frac{99}{2} \\ \mathrm{Thus},\mathrm{Area}=\frac{99}{2}\mathrm{sq}.\mathrm{units} \end{gathered}$$


Hence, the area of the region bounded by the curve y = x² + x, x-axis and lines x = 2 and x = 5 is equal to $\underline{\frac{99}{2}\mathrm{sq}.\mathrm{units}}.$