Question

The area of the region bounded by the curve \( y = \log x \), x-axis and the lines \( x = 1 \), \( x = e \) is

• A. \( \dfrac{1}{e} \) sq. units
• B. \( 1 \) sq. units
• C. \( 4 \) sq. units
• D. \( \dfrac{1}{2} \) sq. units

Hint:

The integral \( \int \log x \, dx = x \log x - x \) is frequently used in problems involving logarithmic curves and areas.

Answer 1

The Correct Option is B

Step 1: Understanding the region.
The given region is bounded by the curve \( y = \log x \), the x-axis, and the vertical lines \( x = 1 \) and \( x = e \). Since \( \log x \geq 0 \) for \( x \geq 1 \), the area can be calculated using definite integration.
Step 2: Set up the integral.
The required area is \[ \text{Area} = \int_{1}^{e} \log x \, dx \] Step 3: Evaluate the integral.
Using the standard result, \[ \int \log x \, dx = x \log x - x \] Step 4: Apply limits.
\[ \left[ x \log x - x \right]_{1}^{e} \] Step 5: Substitute values.
At \( x = e \): \[ e \log e - e = e - e = 0 \] At \( x = 1 \): \[ 1 \log 1 - 1 = -1 \] Step 6: Final calculation.
\[ \text{Area} = 0 - (-1) = 1 \] Step 7: Conclusion.
Hence, the area of the region is \( 1 \) square unit.