#### To determine

**To evaluate:**

∫-31fxdx

#### Answer

∫-31fxdx=6-π4

#### Explanation

**1) Calculation:**

Here the given function is

fx=-x-1 if-3≤x≤0-1-x2 if 0≤x≤1

Now the integral of the function has to be written as a difference of the areas in the following ways

∫-31fxdx

=∫-3-1fxdx+∫-10fxdx+∫01fxdx

=∫-3-1(-x-1)dx+∫-10(-x-1)dx+∫01(-1-x2)dx

=∫-3-1-x-1dx-∫-10x+1dx-∫01(1-x2)dx

Now sketch the graph of each integral as shown below

Then calculate the area of each region

The first integral representsthe black triangle, its base is 2 and height is 2

So its area is =12·base·height=12·2·2=2

The second integral with x+1 represents the green triangle, its base is 1 and height is 1

Therefore, its area =12·base·height=12·1·1=12

Now the integral with 1-x2 represents the quarter part of the circle with radius 1.

Hence its area =πr24=π4

Therefore, the integral

∫-31fxdx=∫-3-1-x-1dx-∫-10x+1dx-∫01(1-x2)dx

=2-12-π4

=6-π4

**Conclusion:**

The value of the integral is ∫-31fxdx=6-π4