Mathematics · Area under the Curves

JEE Main 2026 — 5 April, Morning Shift — Question 39

The area of the region R = {(x,y): xy ≤ 27, 1 ≤ y ≤ x²} is equal to:

Option A:
$78 \log _{e} 3-\frac{52}{3}$
Option B:
$54 \log _{\mathrm{e}} 3-\frac{52}{3}$
Correct
Option C:
$54 \log _{e} 3-\frac{26}{3}$
Option D:
$54 \log _{e} 3+\frac{26}{3}$

Answer: B

Step-by-step solution

Area $=\int_{1}^{3}\left(x^{2}-1\right) d x+\int_{3}^{27}\left(\frac{27}{x}-1\right) d x$ $=\left[\frac{\mathrm{x}^{3}}{3}-\mathrm{x}\right]_{1}^{3}+[27 \ell \mathrm{nx}-\mathrm{x}]_{3}^{27}$ $=\frac{26}{3}-2+27 \ln 9-24$ $=27 \ln 9-\frac{26 \times 2}{3}$ $=54 \ln 3-\frac{52}{3}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 5 April, Morning Shift
Subject: Mathematics
Chapter: Area under the Curves
Topic: Area under the Curves