Class 12 Applications Of Integrals · Very Long Answer5 Marksunbounded_region

Class 12 Maths · Applications Of Integrals · Very Long Answer #43

Determine whether the region {(x,y):x²≤y≤x³} has finite area. If not, establish its divergence.

✅ Step-by-Step Solution

Final Answer: The area is infinite.

A nonempty vertical section requires x²≤x³, or x²(x−1)≥0.
For real x this holds at the isolated input x=0 and for all x≥1. Thus the region extends unboundedly to the right.
The isolated point (0,0) contributes no area; on x≥1 the vertical width is x³−x²≥0.
The area up to x=R>1 is ∫₁ᴿ(x³−x²)dx=R⁴/4−R³/3+1/12. This tends to +∞ as R→∞.
Therefore the stated region has infinite area, not a finite numerical value.

[1 mark] A nonempty vertical section requires x²≤x³, or x²(x−1)≥0.
[1 mark] For real x this holds at the isolated input x=0 and for all x≥1. Thus the region extends unboundedly to the right.
[1 mark] The isolated point (0,0) contributes no area; on x≥1 the vertical width is x³−x²≥0.
[1 mark] The area up to x=R>1 is ∫₁ᴿ(x³−x²)dx=R⁴/4−R³/3+1/12. This tends to +∞ as R→∞.
[1 mark] Therefore the stated region has infinite area, not a finite numerical value.

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