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I was thinking about the following problem:

If for $x \in R$, $\phi (x)$ denotes the integer closest to $x$ (If there are two such integers take the larger one),then $\int_{10}^{12}\phi (x)dx$ equals to

(a)$22$
(b)$11$
(c)$20$
(d)$12$.

I do not know how to proceed with the problem. Could someone point me in the right direction.Thanks in advance for your time.

1 Answers 1

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Rewrite the function $\varphi$ on the interval $[10,12]$ as follows:

$\varphi(x)=\begin{cases} 10,&\text{if }10\le x<10.5\\ 11,&\text{if }10.5\le x<11.5\\ 12,&\text{if }11.5\le x\le 12\;. \end{cases}$

Then

$\int_{10}^{12}\varphi(x)\,dx=\int_{10}^{10.5}10\,dx+\int_{10.5}^{11.5}11\,dx+\int_{11.5}^{12}12\,dx\;,$

which is a straightforward calculation.

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    @user53386: You’re very welco$m$e; I’m glad that it helped.2012-12-18