Mathematics - Definite Integration Question with Solution | TestHub

Here, $R_1={\int }_{-1}^{2}xf(x)dx$ Using, ${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$ $$\begin{array}{rl}& R_1={\int }_{-1}^2(1-x)f(1-x)dx,\\ \therefore & R_1={\int }_{-1}^2(1-x)f(x)dx\dots (1-x)]\end{array}$$ Given, $R_2$ is area bounded by $$\begin{array}{rl}& f(x),x-1\text{ and }x=2\\ \therefore & R_2={\int }_{-1}^{2}f(x)dx\end{array}$$ Adding Eqs. (i) and (ii), we get $$2R_1={\int }_{-1}^{2}f(x)dx$$ $\therefore$ From Eqs. (iii) and (iv), we get $$2R_1=R_2$$