JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Area Under Curves

Area of the region bounded by the curve

y = e^{x}

and lines

x = 0

and

y = e

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

Solution

The region is bounded by

y = e^{x}

x = 0

, and

y = e

. This forms a region from

x = 0

x = 1

(since

e^{1} = e

).

The area can be computed as:

A = \int_{0}^{1} (e - e^{x}) d x = e - (e^{1} - e^{0}) = e - e + 1 = 1

Alternatively, integrating with respect to

y

from

y = 1

y = e

(where

x = ln y

A = \int_{1}^{e} ln y d y = [y ln y - y]_{1}^{e} = (e - e) - (0 - 1) = 1

This confirms option (D). Option (C) gives

e - (e - 1) = 1

✓

For option (B), using the property

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

\int_{1}^{e} ln y d y = \int_{1}^{e} ln (e + 1 - y) d y

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · One or more correct options