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