Answer:
Step-by-step explanation:
To calculate the area bounded by the curves
�
(
�
)
=
�
(
�
−
1
)
f(x)=x(x−1) and
�
(
�
)
=
�
�
g(x)=ex, we need to find the points of intersection between the two curves.
Setting
�
(
�
)
=
�
(
�
)
f(x)=g(x), we have:
�
(
�
−
1
)
=
�
�
x(x−1)=ex
This equation does not have a simple algebraic solution, so we can use numerical methods or graphing software to find the approximate points of intersection. Let's assume for this example that the curves intersect at
�
=
�
x=a and
�
=
�
x=b, where
�
<
�
a<b.
The area bounded by the curves can be calculated as:
�
=
∫
�
�
(
�
�
−
�
(
�
−
1
)
)
�
�
A=∫ab(ex−x(x−1))dx
Integrating term by term, we get:
�
=
[
�
�
−
�
2
2
+
�
2
2
−
�
]
�
�
A=[ex−2x2+2x2−x]ab
�
=
[
�
�
−
�
]
�
�
A=[ex−x]ab
�
=
�
�
−
�
−
(
�
�
−
�
)
A=eb−b−(ea−a)
Therefore, to find the exact area bounded by the curves, we need to first determine the points of intersection between
�
(
�
)
f(x) and
�
(
�
)
g(x).