Question

What is the area under the curve y=x−x^2and above the x-axis?

Answer 1

The area between the x-axis and the given curve equals 1/6 units.

Explanation:

given any 2 functions f(x) and g(x) the area between the 2 figures is calculated as

`A=\int_(x_1)^(x_2)(f(x)-g(x))dx`

The area needed is shown in the attached figure

The points of intersection of the given curve and x-axis are calculated as

`x-x^2=0\\\\x(1-x)=0\\\\\therefore x=0,x=1`

hence the points of intersection are
`(0,0),(1,0)`

The area thus equals

`A=\int_(0)^(1)(x-x^2-0)dx\\\\A=\int_(0)^(1)xdx-\int_(0)^(1)x^2dx\\\\A=1/2-1/3\\\\A=1/6`