ISI2014-DCG-20

Question

If $A(t)$ is the area of the region bounded by the curve $y=e^{-\mid x \mid}$ and the portion of the $x$-axis between $-t$ and $t$, then $\underset{t \to \infty}{\lim} A(t)$ equals

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

According to the question,
$\displaystyle A(t)=\int_{-t}^{t} e^{-|x|}\mathrm{d}x$

Since $|x|=\left\{\begin{matrix} -x & ; x<0\\ x & ; x\ge0 \end{matrix}\right.$

Hence $\displaystyle \int_{0}^{t} e^{-|x|}\mathrm{d}x=\int_{0}^{t} e^{-x}\mathrm{d}x  \tag{i}$

and

$\displaystyle \int_{-t}^{0} e^{-|x|}\mathrm{d}x=\int_{-t}^{0} e^{-(-x)}\mathrm{d}x=\int_{-t}^{0} e^{x}\mathrm{d}x \tag{ii}$

Here $x \in [-t,0)\Rightarrow x<0$.

Let $x=-y\Rightarrow \mathrm{d}x=-\mathrm{d}y$.

When $x=-t$, then $y=t$ and when $x=0$, then $y=0$

$\begin{align} \displaystyle \therefore \int_{-t}^{0} e^{x}\mathrm{d}x&=\int_{t}^{0} e^{-y}(-\mathrm{d}y)\\ \displaystyle &=-\int_{t}^{0} e^{-y}\mathrm{d}y=\int_{0}^{t} e^{-y}\mathrm{d}y ~;~ \left[\because \int_{a}^{b}f(x)\mathrm{d}x=- \int_{b}^{a}f(x)\mathrm{d}x \right] \tag{iii} \end{align}$

 

Look at the integral $\displaystyle \int_{0}^{t} e^{-y}\mathrm{d}y$. Here $y$ is just a variable. If we write $\displaystyle \int_{0}^{t} e^{-y}\mathrm{d}y= \displaystyle \int_{0}^{t} e^{-v}\mathrm{d}v$. It's also valid. Even so we can write $\displaystyle \int_{0}^{t} e^{-y}\mathrm{d}y= \displaystyle \int_{0}^{t} e^{-x}\mathrm{d}x$. 

$\begin{align} \therefore \int_{-t}^{0} e^{x}\mathrm{d}x &=\int_{0}^{t} e^{-x}\mathrm{d}x ~;~[\mathrm{From~no(iii)}] \\ \Rightarrow  \int_{-t}^{0} e^{-|x|}\mathrm{d}x &=\int_{0}^{t} e^{-x}\mathrm{d}x ~;~[\mathrm{From~no(ii)}] \tag{iv} \end{align}$

 

 

$\begin{align} \therefore A(t)=\int_{-t}^{t} e^{-|x|}\mathrm{d}x&=\int_{-t}^{0} e^{-|x|}\mathrm{d}x+\int_{0}^{t} e^{-|x|}\mathrm{d}x\\  &=\int_{0}^{t} e^{-x}\mathrm{d}x+\int_{0}^{t} e^{-x}\mathrm{d}x ~;~~[\mathrm{Using~no(iv)~and~no(i)}] \\  &=2\int_{0}^{t} e^{-x}\mathrm{d}x \\ &=-2\left[ e^{-x}\right]_{0}^{t}\\ &=-2\left( e^{-t}-e^0\right)=2 \left( 1-\frac{1}{e^t} \right) \end{align}$

 

$$\because t \to \infty \Rightarrow \frac{1}{e^t}\to 0 \\ \displaystyle \therefore \lim_{t \to \infty} A(t) = \lim_{t \to \infty} 2 \left( 1-\frac{1}{e^t} \right)= 2$$.

The correct answer is C.

Shortcut Idea: It's easily intuitive that $y$-axis cuts $e^{-|x|}$ graph into half. See it below. Then evaluating the area of the right side ($x\ge0$) of the graph i.e. $\displaystyle \int_{0}^{t}e^{-x}\mathrm{d}x$ is just enough.

 

So, $A(t)$ will be $\displaystyle 2\times \int_{0}^{t}e^{-x}\mathrm{d}x=2\left(1-\frac{1}{e^t} \right)$

 

So the correct answer is C.