Question

The area bounded by the curve y = log_e x and x-axis and the straight line x = e is ___________ .

Options

• e sq. units

• 1 sq. units

• 1−\[\frac{1}{e}\] sq. units

• 1+\[\frac{1}{e}\] sq. units

Answer 1

1 sq. units

 

The point of intersection of the curve and the straight line is A(e, 1).
Therefore, the area of the required region ABC,
\[A = \int_0^1 \left( x_1 - x_2 \right) d y ..........\left(\text{ where, }x_1 = e\text{ and }x_2 = e^y \right)\]
\[ = \int_0^1 \left( e - e^y \right) d y\]
\[ = \left[ ey - e^y \right]_0^1 \]
\[ = \left\{ e\left( 1 \right) - e^\left( 1 \right) \right\} - \left\{ e\left( 0 \right) - e^\left( 0 \right) \right\}\]
\[ = e - e + 1\]
\[ = 1\text{ square unit }\]