Find the area bounded by the curve y=xe^x and its asymptotes.,

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Title: Find the area bounded by the curve y=xe^x and its asymptotes.

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Posted On: Jan 07, 2012 10:11 AM

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Shubham Satyam
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Jan 10, 2012 10:11 AM

$\hspace{-16}$Clearly $\mathbf{y=0}$ is a Asymptotes of the Curve $\mathbf{f(x)=xe^x}$\\\\ So Required Area is $\mathbf{A=\int_{-\infty}^{0}x.e^xdx}$\\\\\\ Now Let $\mathbf{k\rightarrow -\infty}$, Then $\mathbf{A=\lim_{k\rightarrow -\infty}\int_{k}^{0}x.e^xdx}$\\\\ Using Integration By parts\;, We Get\\\\ $\mathbf{A= \lim_{k\rightarrow -\infty}\int_{k}^{0}\left[e^x(x-1)\right]_{k}^{0}=\lim_{k\rightarrow -\infty}ke^k-\lim_{k\rightarrow -\infty}e^k-0-e^0}$\\\\\\ Now $\mathbf{\lim_{k\rightarrow -\infty}k.e^k}$\\\\ Put $\mathbf{k=t}$ and $\mathbf{t\rightarrow 0}$\\\\ So $\mathbf{\lim_{k\rightarrow -\infty}k.e^k=\lim_{k\rightarrow 0}\frac{k}{e^k}=0}$(Using L, Hopital Rule)\\\\ So $\mathbf{A=0-0-0-1}$\\\\ So Area $\mathbf{A=\mid -1 \mid = 1}$\\\\$

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