The area of the region bounded by the curves y=ex, text y=logex and lines x=1, text x=2 is

Question

The area of the region bounded by the curves y=ex, text y=logex and lines x=1, text x=2 is (A)  (e-1)2  (B)  e2-e+1  (C)  e2-e+1-2loge2  (D)  e2+e-2loge2

Tardigrade Admin · Accepted Answer

Required area, A=∫12(ex- log ex) dx =[ex]12-[x log ex-∫1 dx]12 =e2-e-[x log ex-x]12 =e2-e-[2 log e2-2(0-1)] =e2-e-2 log e2+1

Q. The area of the region bounded by the curves $y = e x, y = l o g_{e} x$ and lines $x = 1, x = 2$ is

$(e - 1)^{2}$

$e^{2} - e + 1$

$e^{2} - e + 1 - 2 l o g_{e} 2$

$e^{2} + e - 2 l o g_{e} 2$

Required area, $A = \int_{1}^{2} (e^{x} - lo g_{e} x) d x$
$= [e^{x}]_{1}^{2} - [x lo g_{e} x - \int 1 d x]_{1}^{2}$
$= e^{2} - e - [x lo g_{e} x - x]_{1}^{2}$
$= e^{2} - e - [2 lo g_{e} 2 - 2 (0 - 1)]$
$= e^{2} - e - 2 lo g_{e} 2 + 1$

Q. The area of the region bounded by the curves y=ex, y=loge​x and lines x=1,x=2 is

(e−1)2

e2−e+1

e2−e+1−2loge​2

e2+e−2loge​2