Definite Integration
Grade None
Question:
<p>Let d<sub>1</sub> [(x<sub>1</sub>, y<sub>1</sub>), (x<sub>2</sub>, y<sub>2</sub>)] = |x<sub>1</sub> - x<sub>2</sub>| + |y<sub>1</sub> - y<sub>2</sub>| and d<sub>2</sub> [(x<sub>1</sub>, y<sub>1</sub>), (x<sub>2</sub>, y<sub>2</sub>)] = <span class="math-tex">\(\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}\)</span> denote the distance between (x<sub>1</sub>, y<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>) on the coordinate plane. The area of the region enclosed by the set of points (x, y) satisfying d<sub>1</sub> [(x, y), (0, 0)] <span class="math-tex">\(\geq\)</span> 1 and d<sub>2</sub> [(x, y), (0, 0)] <span class="math-tex">\(\leq\)</span> 1, is :</p>
<p style="display:inline"><span class="math-tex">\(\pi+2\)</span></p>
<p style="display:inline"><span class="math-tex">\(\pi-4\)</span></p>
<p style="display:inline"><span class="math-tex">\(\pi+4\)</span></p>
<p style="display:inline"><span class="math-tex">\(\pi-2\)</span></p>
Step-by-Step Solution
Key Concept: Identify the region as the area inside a unit circle ($x^2 + y^2 \leq 1$) and outside a square defined by the Manhattan distance ($|x| + |y| \geq 1$).
<p><span class="math-tex">$\pi-2$</span></p>
Correct Answer: D