#### 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≈-1.891 and x≈1.135

ii) Area of the region bounded by the curves

A≈6.31

#### Explanation

**1) Concept:**

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=x-cosx, y=2-x2

3) **Calculation:**

fx=x-cosx and gx=2-x2

i) To find the intersection points of the curves draw the graph.

From the sketch, it is clear that the curves intersect at x≈-1.189, x≈1.135

In the interval -1.189, 1.135

ii) the upper curve is y=2-x2 and the lower curve is y=x-cosx

A=∫-1.1891.135fx-g(x)dx

=∫-1.1891.1352-x2-(x-cosx) dx

=∫-1.1891.1352-x2-x+cosxdx

=2x-13x3-12x2+sinx1.135-1.189

Plugging the limit values,

=2(1.135)-13(1.135)3-12(1.135)2+sin(1.135)-2(-1.189)-13(-1.189)3-12(-1.189)2+sin(-1.189)

A≈6.31

**Conclusion:**

i) Intersection points of the curves:

x≈-1.189, x≈1.135

ii) Area of the region bounded by the curves:

A≈6.31