Question

Area of the region bounded by the curve $ y = e^x $ and lines
$x = 0$ and $y = e$ is

• 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 $

Answer 1

Answer: (B), (C), (D)

Shaded area $ = e - \bigg(\int \limits_0^1 e^x dx \bigg) = 1 $
Also, $\, \, \, \int \limits_1^e ln\, \, (e + 1 - y) dy\, \, [put\, \, e+ 1 - y = t \Rightarrow - dy = dt]$
$\, \, \, =\int \limits_e^1 ln\, \, t \, \, (-dt)$ $ = \int \limits_1^e ln\, \, t \, \, dt $ $ = \int \limits_1^e ln\, \, y \, \, dy = 1 $