Section B : Numerical Type Questions.

Question

Section B : Numerical Type Questions. Let $[t]$ denote the greatest integer $\le t$. Then the value of $8\displaystyle\int_{-1}^{1/2}([2x]+|x|)\,dx$ is

Answer: 5

Accepted Answer

Split according to the value of $[2x]$: for $-1\le x<-\tfrac12$, $[2x]=-2$; for $-\tfrac12\le x<0$, $[2x]=-1$; for $0\le x<\tfrac12$, $[2x]=0$.

$$I=\int_{-1}^{-1/2}(-2+|x|)dx+\int_{-1/2}^{0}(-1+|x|)dx+\int_{0}^{1/2}(0+|x|)dx$$

Evaluating each piece (using $|x|=-x$ for $x<0$) and summing gives $I=\dfrac58$.

$$\therefore 8I=8\cdot\frac58=5$$

Solutions · 1