Question

The area bounded by the curve y = ln (x) and the lines y = 0, y = ln (3) and x = 0 is equal to :

• A. 3
• B. 3 ln (3) - 2
• C. 3 ln (3) + 2
• D. 2

Answer 1

Answer: (D)

To find the point of intersection of curves y = ln (x)
and y = ln (3), put ln (x) = ln (3)
$\Rightarrow$ ln (x) - ln (3) = 0
$\Rightarrow$ ln (x) - ln (3) = ln (1)
$\Rightarrow \frac{x}{3}=1,\Rightarrow x=3$

Required area $=\underset{0}{\overset{3}{\int }}\text{ln}\left(3\right)dx-\underset{1}{\overset{3}{\int }}\text{ln}\left(x\right)dx$
$={\left[x\text{ln}\left(3\right)\right]}_0^3-{\left[x\text{ln}\left(x\right)-x\right]}_1^3=2$