#### To determine

**To evaluate:**

The integral ∫-4312xdx by interpreting it in terms of areas.

#### Answer

254

#### Explanation

**1) Concept:**

A definite integral can be interpreted as a net area, that is, 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:**

∫-4312xdx

**4) Calculation:**

The given integral ∫-4312xdx can be interpreted as the area under the graph of

fx= 12x between x= -4 and x=3

The graph of fx= 12x is shown below

Now, find the integral as the net area of the two triangles

By using the formula of the area of a triangle,

A1=12bh=12*4*2=4

A2=12bh=12*3*1.5=94

Thus, the value of the original integral is (both regions are above x-axis)

∫-4312xdx=A1+A2=4+94=254

**Conclusion:**

∫-4312xdx=254