#### To determine

**a)**

**To find:**

The total area between the curves for 0≤x≤5

#### Answer

39

#### Explanation

**1) Concept:**

Formula:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is

A= ∫abfx-gxdx

fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)

**2) Given:**

3) **Calculation:**

To find the area between the curves for 0≤x≤5,

From the given diagram, for 0≤x≤2, A=12

And for 2≤x≤5, A=27

Adding both the areas together,

12+27=39

**Conclusion:**

The total area between the curves for 0≤x≤5=39

#### To determine

**b)**

**To find:**

The value of ∫05fx-g(x) dx

#### Answer

15

#### Explanation

**1) Concept:**

Formula:

The area A of region bounded by curves y=f(x), y=g(x) and the lines x=a and x=b is

A= ∫abfx-gxdx

fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)

**2) Given:**

3) **Calculation:**

From the given diagram, it is clear that the curves are intersecting each other at x=2.

Therfore, the interval 0≤x≤5 should be divided into two sub intervals,

0≤x≤2 and 2≤x≤5

Therefore,

∫05fx-g(x) dx= ∫02fx-g(x) dx+∫25fx-g(x) dx

For 0≤x≤2, the curve g is the upper curve, and the curve f is the lower curve.

So the area between g and f when 0≤x≤2 is given by ∫02gx-f(x) dx. But ∫02gx-f(x) dx=12. So ∫05fx-g(x) dx= ∫02fx-g(x) dx+∫25fx-g(x) dx

= ∫02-gx-f(x) dx+∫25fx-g(x) dx

=-12+27

=15

**Conclusion:**

The value of ∫05fx-g(x) dx=15