Question

The area bounded by the curve \( x = \log(|y|) \), the lines \( x = -1 \) and \( x = 0 \) is:

• A. \( 1 - e^{-1} \)
• B. \( 1 - e \)
• C. \( 2(1 - e) \)
• D. \( 2(1 - e^{-1}) \)

Hint:

When the function involves \( |y| \), consider symmetry in area calculations. \( \log|y| \) leads to positive and negative branches.

Answer 1

The Correct Option is D

We are given the curve \( x = \log |y| \). Solving for \( y \), we get \( y = \pm e^x \). This represents two branches: one in the positive \( y \)-axis and one in the negative, symmetric about the \( x \)-axis. To find the total area between the lines \( x = -1 \) and \( x = 0 \), we compute: \[ \text{Area} = \int_{-1}^{0} (e^x - (-e^x)) \, dx = \int_{-1}^{0} 2e^x \, dx \] \[ = 2 \int_{-1}^{0} e^x \, dx = 2 \left[ e^x \right]_{-1}^{0} = 2(e^0 - e^{-1}) = 2(1 - e^{-1}) \]