QJEEAdv19Ma3123: JEE Advanced 2019 Mathematics - Class 12 JEEAdv19Ma3123 Solution: The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x 2 } is

Question

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The area of the region

{(x, y) : xy

\leq

8, 1

\leq

y

\leq

x²} is

Accepted Answer

The given region

{(x, y) : xy

\leq

8, 1

\leq

y

\leq

x²}.

From the figure, region A and B satisfy the given region, but only A is bounded region, so area of bounded region

A = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x

[

∴

Points P(1, 1), Q(2, 4) and R(8, 1)]

= {[\frac{x^{3}}{3} - x]}_{1}^{2} + [8 \log | x | - x]_{2}^{8}

= (\frac{8}{3} - 2 - \frac{1}{3} + 1) + 8 \log 8 - 8 - 8 \log 2 + 2

= - \frac{14}{3} + 16 \log 2

= 16 \log 2 - \frac{14}{3}

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