Area Under the Curve
Area between logarithmic and exponential curves
Grade 12
Question:
<p>The area enclosed between the curves \(y = \log_e(x + e)\), \(x = \log_e\left(\frac{1}{y}\right)\) and the x-axis is</p>
<p>(A) 2</p>
<p>(B) 1</p>
<p>(C) 4</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Convert logarithmic conditions to exponential form to identify the curves; find intersection and integrate.
<p>The curve $y = \log_e(x + e)$ passes through certain points. The condition $x = \log_e\left(\frac{1}{y}\right)$ is equivalent to $e^x = \frac{1}{y}$ or $y = e^{-x}$. Find intersection points and integrate to obtain area = 1.</p>
Correct Answer: B