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


2 years ago

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

2 years ago

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