A region in the xy-plane satisfies x≥0, x+y≤3, x²≤4y and y≤1+√x. If its area is k square units, find 10k.
Physics · Mixed Chapter
JEE Main Practice

Question

A region in the xy-plane satisfies x≥0, x+y≤3, x²≤4y and y≤1+√x. If its area is k square units, find 10k.

G1 Physics · Accepted Answer

Step 1: Express lower boundary y=x²/4 and upper boundary y=min(3−x,1+√x). Step 2: Find intersection ranges with x≥0 and the requirement x²/4≤upper boundary. Step 3: The upper curves meet at x=1, while the lower parabola meets y=3−x at x=2. Thus k=∫₀¹(1+√x−x²/4)dx +∫₁²(3−x−x²/4)dx. Step 4: First integral =19/12, second =11/12; total area k=30/12=5/2. Thus 10k=25, NOT source key 4.