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Joint Entrance Examination
Mathematics
Area Under Curves
Question 41
Question 41 of 230
Medium
MCQ
The area of the region
(
x
,
y
)
:
x
2
≤
y
≤
∣
x
2
−
4
∣
,
y
≥
1
{(x,y):{x}^{2}\le y\le |{x}^{2}-4|,y\ge 1}
(
x
,
y
)
:
x
2
≤
y
≤
∣
x
2
−
4∣
,
y
≥
1
is
A
4
3
(
4
2
+
1
)
\frac{4}{3}(4\sqrt{\sqrt{2}}+1)
3
4
(
4
2
+
1
)
B
3
4
(
4
2
+
1
)
\frac{3}{4}(4\sqrt{\sqrt{2}}+1)
4
3
(
4
2
+
1
)
C
4
3
(
4
2
−
1
)
\frac{4}{3}(4\sqrt{\sqrt{2}}-1)
3
4
(
4
2
−
1
)
D
3
4
(
4
2
−
1
)
\frac{3}{4}(4\sqrt{\sqrt{2}}-1)
4
3
(
4
2
−
1
)
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Official solution
\[
Required area
=
2
[
∫
1
2
y
d
y
+
∫
4
2
4
−
y
d
y
]
=
2
[
y
3
/
2
3
2
]
1
2
−
2
(
4
−
y
)
3
/
2
3
]
2
4
]
=
4
3
(
4
2
−
1
)
\begin{aligned} & \text { Required area }=2\left[\int_1^2 \sqrt{y} d y+\int_4^2 \sqrt{4-y} d y\right] \\\\ & \left.\left.=2\left[\frac{y^{3 / 2}}{\frac{3}{2}}\right]_1^2-\frac{2(4-y)^{3 / 2}}{3}\right]_2^4\right] \\\\ & =\frac{4}{3}(4 \sqrt{2}-1) \end{aligned}
Required area
=
2
[
∫
1
2
y
d
y
+
∫
4
2
4
−
y
d
y
]
=
2
[
2
3
y
3/2
]
1
2
−
3
2
(
4
−
y
)
3/2
]
2
4
=
3
4
(
4
2
−
1
)
\]
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