#### To determine

**To evaluate:**

The integral ∫-12(1-x)dx by interpreting it in terms of areas.

#### Answer

32

#### Explanation

**1) Concept:**

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

∫abf(x)dx=A1-A2

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:**

A=12bh

where b is the base and h is the height

**3) Given: **

∫-12(1-x)dx

**4) Calculation:**

∫-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.

By using the formula of the area of a triangle,

A2=12*1*1=12

So, we have

∫-12(1-x)dx=A1-A2=2-12 =32

Therefore,

∫-12(1-x)dx =32

**Conclusion:**

∫-12(1-x)dx=32