Question

the area of the region boundeby the curves y³=x² and y=2-x​

Answer 1

Answer:

Answer

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: