#### 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=0 and x≈1.052

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

A≈0.59

#### 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=xx2+12, y=x5-x, x≥0

3) **Calculation:**

fx=xx2+12 and gx=x5-x

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

From the sketch, it is clear that the curves intersect at x=0 and x=1.052

ii) the upper curve is y=xx2+12 and the lower curve is y=x5-x

Therefore,

A=∫01.052xx2+12-(x5-x)dx

=∫01.052xx2+12-x5+xdx

Applying integration separately,

=∫01.052xx2+12dx-∫01.052x5dx+∫01.052xdx …(1)

Now solving,

∫01.052xx2+12dx

Substitute u=x2+1 so, du=2xdx⇒xdx=du2

When x=0, u=1 and when x=1.052, u=2.106704

Therefore,

∫01.052xx2+12dx=∫12.10670412u2du

=12∫12.106704u-2du

=12u-2+1-2+112.106704

=-121u12.106704

Substitute value of u,

=-121x2+101.052

Now, from equation (1),

A=-121x2+101.052-∫01.052x5dx+∫01.052xdx

By using power rule of Integration,

=-121x2+1-x66+x2201.052

=87500875148281332

A=0.59010041

That is,

A≈0.59

**Conclusion:**

i) The intersection points of the curves:

x=0 and x≈1.052

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

A≈0.59