The area (in sq. units) bounded between the curves \( x=e^{y} \), \( x=e \) and \( x \)-axis in the first quadrant is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
The curve \( x=e^y \) is \( y=\log x \). It meets the x-axis at x = 1 and the line x = e at y = 1. The region lies under y = log x from x = 1 to x = e:
\( \displaystyle \text{Area}=\int_1^e\log x\,dx \)
Integrating by parts:
\( \displaystyle \int_1^e\log x\,dx=\Big[x\log x-x\Big]_1^e \)
\( \displaystyle =(e-e)-(0-1)=1 \)
Shortcut: Use horizontal strips: for 0 ≤ y ≤ 1 the strip runs from \( x=e^y \) to x = e, so the area is
\( \displaystyle \int_0^1(e-e^y)\,dy=e-(e-1)=1 \)
with no integration by parts.
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1-e}{e} \) — Incorrect. \( \frac{1-e}{e} \) is negative, so it cannot be an area.
Option (2) \( \displaystyle\frac{e-1}{e} \) — Incorrect. \( \frac{e-1}{e}\approx 0.63 \) is \( \int_0^1 e^{-y}\,dy \), a different curve.
Option (4) \( e-1 \) — Incorrect. \( e-1=\int_0^1 e^y\,dy \) is the area to the left of the curve (between the curve and the y-axis), not the region bounded by x = e.