IIT-JEE 1994Question 1
In what ratio does the $x$-axis divide the area of the region bounded by the parabolas $y=4x-x^{2}$ and $y=x^{2}-x$?
Application of Integrals
Answer$121:4$
Solution
The parabolas meet at $x=0$ and $x=\frac52$; total area $=\int_0^{5/2}(5x-2x^2)dx=\frac{125}{24}$. Below the $x$-axis lies the part of $y=x^2-x$ on $[0,1]$ with area $\frac16=\frac4{24}$. The rest is $\frac{121}{24}$, so the ratio is $121:4$.