MCQ
The area bounded by the curves $y = {\log _e}x$ and $y = {({\log _e}x)^2}$ is
- ✓$3 - e$
- B$e - 3$
- C$\frac{1}{2}(3 - e)$
- D$\frac{1}{2}(e - 3)$
✓
Answer
Correct option: A.
$3 - e$
a
(a) Required area $ = \int_1^e {[\log x - {{(\log x)}^2}]} \,dx$
(a) Required area $ = \int_1^e {[\log x - {{(\log x)}^2}]} \,dx$
$A = \int_1^e {\,\log x\,dx} - \int_1^e {{{(\log x)}^2}dx} $
$ = [x\log x - x]\,_1^e - [x{(\log x)^2} - 2x\log x + 2x]\,_1^e$
$ = [e - e - ( - 1)] - [e{(1)^2} - 2e + 2e - (2)]$
$ = (1) - (e - 2)$$ = 3 - e$.
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