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Byju's Answer
Standard XI
Mathematics
Area between Two Curves
The area boun...
Question
The area bounded by the curves
y
=
x
e
x
,
y
=
x
e
−
x
and the line
x
=
1
is
A
2
e
sq. units
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B
1
−
2
e
sq. units
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C
1
e
sq. units
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D
1
−
1
e
sq. units
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Solution
The correct option is
A
2
e
sq. units
y
=
x
e
x
Let
d
y
d
x
=
0
⇒
e
x
+
x
e
x
=
0
⇒
x
=
−
1
Also, at
x
=
−
1
,
d
y
d
x
changes sign from
−
v
e
to
+
v
e
.
Hence,
x
=
−
1
is a point of minima.
When
x
→
∞
,
y
→
∞
Also,
lim
x
→
−
∞
x
e
x
=
lim
x
→
−
∞
x
e
−
x
=
lim
x
→
−
∞
1
−
e
−
x
(Applying L'Hospital Rule)
=
0
Similarly, we can draw the graph of
y
=
x
e
−
x
.
Required area
=
1
∫
0
x
e
x
d
x
−
1
∫
0
x
e
−
x
d
x
=
[
x
e
x
]
1
0
−
1
∫
0
e
x
d
x
−
⎛
⎜
⎝
[
−
x
e
−
x
]
1
0
+
1
∫
0
e
−
x
d
x
⎞
⎟
⎠
=
e
−
(
e
−
1
)
−
(
−
e
−
1
−
(
e
−
1
−
1
)
)
=
2
e
sq. units.
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