The integral ∫-12(1-x)dx by interpreting it in terms of areas.
A definite integral can be interpreted as a net area, that is, as difference of areas;
where A1 is the region above x- axis and below the graph of , and A2 is the region below x- axis and above the graph of f.
2) Formula for Area of a triangle:
where b is the base and h is the height
∫-12(1-x)dx this given integral can be interpreted as the area under the graph of
fx= 1-x between x= -1 and x=2
The graph of y= 1-x is the line with the slope -1 as shown in the figure below.
Now, find the integral as the difference of the areas of the two triangles.
A1 is the region above x- axis and below the graph of f.
By using the formula of the area of a triangle,
A1 = 12*2*2=2
A2 is the region below x-axis and above the graph of f.
So, we have