Question

Area enclosed by \[ x^2 + 4y^2 \leq 4, \quad |x| \leq 1, \quad y \geq 1 - |x| \text{ is} \]

• A. \( 4 \sin^{-1} \left( \frac{3}{5} \right) + \frac{6}{5} \)
• B. \( \sin^{-1} \left( \frac{3}{5} \right) + \frac{6}{5} \)
• C. \( 4 \sin^{-1} \left( \frac{3}{5} \right) + \frac{12}{5} \)
• D. \( 4 \sin^{-1} \left( \frac{3}{5} \right) - \frac{6}{5} \)

Hint:

For problems involving bounded areas, carefully interpret the inequalities and set up integrals or geometric relations that match the given constraints.

Answer 1

The Correct Option is D

Step 1: Interpret the inequalities.
We are given the inequalities: \[ x^2 + 4y^2 \leq 4, \quad |x| \leq 1, \quad y \geq 1 - |x| \] The first inequality represents an ellipse, the second bounds \( x \), and the third defines the region for \( y \). Step 2: Set up the area integral.
The area enclosed by these inequalities can be found by integrating over the region. First, convert the ellipse equation into the standard form and adjust the bounds for \( x \) and \( y \). Step 3: Compute the area.
Using the given constraints, we calculate the area of the region. The result involves the inverse sine function, leading to the area expression: \[ \text{Area} = 4 \sin^{-1} \left( \frac{3}{5} \right) - \frac{6}{5} \]