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Joint Entrance Examination
Mathematics
Differential Equations
Question 110
Question 110 of 490
Medium
MCQ
The area of the region
R
=
{
(
x
,
y
)
:
x
y
≤
27
,
1
≤
y
≤
x
2
}
R=\{(x,y): xy\le27,\ 1\le y\le x^2\}
R
=
{(
x
,
y
)
:
x
y
≤
27
,
1
≤
y
≤
x
2
}
is:
A
78
log
e
3
−
52
3
78\log_e3-\dfrac{52}{3}
78
lo
g
e
3
−
3
52
B
54
log
e
3
−
52
3
54\log_e3-\dfrac{52}{3}
54
lo
g
e
3
−
3
52
C
54
log
e
3
−
26
3
54\log_e3-\dfrac{26}{3}
54
lo
g
e
3
−
3
26
D
54
log
e
3
+
26
3
54\log_e3+\dfrac{26}{3}
54
lo
g
e
3
+
3
26
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Official solution
\[\text{Area} = \int_1^3(x^2-1)\,dx+\int_3^{27}\left(\frac{27}{x}-1\right)dx = \left[\frac{x^3}{3}-x\right]_1^3+[27\ln x-x]_3^{27}\] \[= \frac{26}{3}-2+27\ln9-24 = 54\ln3-\frac{52}{3}.\]
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