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

Hint:

Integrate with respect to \(y\): for each \(y\) between 0 and \(\ln 3\), \(x\) runs from 0 to \(e^y\).

Answer 1

The Correct Option is D

The curve gives \(x = e^{y}\), and the region lies between \(x = 0\) and \(x = e^{y}\) for \(0 \le y \le \ln 3\).
Area \(= \int_0^{\ln 3} e^{y}\,dy = \left[e^{y}\right]_0^{\ln 3}\).
\(= e^{\ln 3} - e^{0} = 3 - 1 = 2\).
Answer: (4) 2