Question

Find the area or the region bounded by the curves y = x^3 and y = x.

Answer 1

wgiven the functions

`y=x^3`
`y=x`

the area of the curve is given by the integral

the limits are defined by

`y1=y2`
`x^3=x`
`x^3-x=0`
`x(x+1)(x-1)=0`
`x=0;x=-1;x=1`

the defined integral for the ara is given by

`A=-\int_(-1)^0x-x^3dx+\int_0^1x-x^3dx`
`A=2\int_0^1x-x^3dx`
`A=2\lbrack(x^2)/(2)-(x^4)/(4)\rbrack_0^1`
`A=2\lbrack(1^2)/(2)-(1^4)/(4)\rbrack_^-2\lbrack(0^2)/(2)-(0^4)/(4)\rbrack`
`A=2\lbrack(1^2)/(2)-(1^4)/(4)\rbrack_^`
`A=2*(1)/(4)`
`A=(2)/(4)=(1)/(2)`

A= 1/2= 0.5