question_answer

Which of the following is not the area of the region bounded by \[y={{e}^{x}}\] and \[x=0\] and y = e?

A) \[e-1\]

B) \[\int\limits_{1}^{e}{ln(e+1-y)dy}\]

C) \[e-\int\limits_{0}^{1}{{{e}^{x}}dx}\]

D) \[\int\limits_{1}^{e}{\,ln\,\,y\,\,dy}\]

Correct Answer: C

Solution :

[c] Required area \[=\int\limits_{1}^{e}{ln\,\,y\,\,dy}\] \[=(y\,\,ln\,\,y-y)_{1}^{e}=(e-e)-[-1]=1\] Also, \[\int\limits_{1}^{e}{ln\,\,ydy=\int\limits_{1}^{e}{ln(e+1-y)dy}}\] Further, required area \[=e\times 1-\int\limits_{0}^{1}{{{e}^{x}}\,\,dx}\]