#### To determine

**To find:**

Approximate x-coordinates of the points of intersection of the given curves, and find the area of the region bounded by the curves.

#### Answer

i) x-coordinates of the points of intersection of the given curves:

x≈-1.11 and x≈1.25 and x≈2.86

ii) Area of the region bounded by the curves:

A≈8.38

#### 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:**

y=3x2-2x, y=x3-3x+4

3) **Calculation:**

fx=3x2-2x and gx=x3-3x+4

i) To find the intersection point of the curves, draw the graph.

From the sketch, it is clear that the curves intersect at x≈-1.11, x≈1.25 and x≈2.86

in the interval [-1.11, 1.25] the upper curve is y=x3-3x+4 and the lower curve is y=3x2-2x

and in the interval 1.25, 2.86, the upper curve is y= 3x2-2x and the lower curve is y=x3-3x+4

Therefore,

A=∫-1.112.86|fx-g(x)|dx

=∫-1.111.25[x3-3x+4-3x2-2x] dx+∫1.252.863x2-2x-x3-3x+4dx

=∫-1.111.25x3-3x2-x+4dx+∫1.252.86(-x3+3x2+x-4)dx

=14x4-x3-12x2+4x1.25-1.11+-14x4+x3+12x2-4x2.861.25

A≈8.38

**Conclusion: **

i) The intersection points of the curves:

x≈-1.11, x≈1.25 and x≈2.86

ii) The area of the region bounded by the curves:

A≈8.38