the area of the region boundeby the curves y³=x² and y=2-x
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Given curve is y=2x−x
2
−y=x
2
−2x
−y+1=x
2
−2x+1
−(y−1)=(x−1)
2
Which represents a downward parabola with vertex at (1,1)
Point of intersection of the parabola and the line y=x
Put y=x
−(x−1)=(x−1)
2
−x+1=x
2
−2x+1
⇒x
2
−x=0
x(x−1)=0
⇒x=0,1
∴ Points of intersections are (0, 0) and (1, 1).
∴ The area enclosed between the curve y=2x−x
2
and the line y = x
∫
0
1
(2x−x
2
−x)dx=∫
0
1
(x−x
2
)dx=[
2
x
2
−
3
x
3
]
0
1
=(
2
1
−
3
1
)−(0−0)=
6
1
sq. unit.
Step-by-step explanation:
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