Using integration find the area of each of the enclosed regions

Question

1) x-axis and f(x)=2x-x^2

 

2) f(x)=x^2 and g(x)=x

 

 

3) f(x)=x^3-3x and g(x)=x

Answer 1

a--------

 

f(x)=2x-x^2

2x-x^2=0

gives x=0. ,  x=2

 

Integral (2x-x^2). With limits 0 to 2

2(x^2)/2-x^3/3

Plugging in limits

 

=4-(8/3)=4/3

 

Area=4/3.

 

b----------

 

Point of intersections=

2x=x^2

x=0.  x=2

 

Integral of (2x-x^2) with limits as 0 to 2

 

Same as above =4/3

 

c----------

 

Likewise do part c yourself

 

You shall get the same answer