IIT-JEE 1995 ScreeningQuestion 1
The value of $\displaystyle\int_\pi^{2\pi}[2\sin x]\,dx$ where $[\,.\,]$ represents the greatest integer function is
Definite Integration
Solution
Concept. On $(\pi,2\pi)$, $2\sin x\in[-2,0)$, so $[2\sin x]$ is $-1$ or $-2$. It is $-2$ exactly where $2\sin x<-1$. Formulas used. (1) $\sin x=-\frac12$ at $x=\frac{7\pi}6,\frac{11\pi}6$; $\sin x<-\frac12$ between themStep 1 - Pieces. By (1) (end points do not matter): $(\pi,\frac{7\pi}6]$ and $[\frac{11\pi}6,2\pi)$: value $-1$, total length $\frac\pi3$. $(\frac{7\pi}6,\frac{11\pi}6)$: value $-2$, length $\frac{2\pi}3$.Step 2 - Add the rectangles. $$-1\cdot\frac\pi3-2\cdot\frac{2\pi}3=-\frac{5\pi}3.$$ Answer: (A)