To determine

To evaluate:

The integral ∫012x-1dx by interpreting it in terms of areas.

Answer

12

Explanation

1) Concept:

A definite integral can be interpreted as a net area, that is, as a difference of areas;

∫abf(x)dx=A1-A2

where A1 is the region above x- axis and below the graph of f, and A2 is the region below x- axis and above the graph of f.

2) Formula:

Area of a triangle : A=12bh

where b is the base and  h   is the height.

3) Given:

∫012x-1dx

4) Calculation:

The given integral ∫012x-1dx can be interpreted as the area under the graph of

fx= | 2x-1|  between x= 0 and x=1 that means sum of the areas of the two shaded triangles.

Now  the integral is net area, but the region is above x-axis so we may add the area of two triangles.

By using formula of area of a triangle,

A1=12bh=12*0.5*1=0.52=14

A2=12bh=12*0.5*1=0.52=14

Therefore, the net area is

∫-012x-1dx=A1+A2=0.52+0.52=12

Conclusion:

∫012x-1dx=12