#### 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) The intersection points of the curves:

x=0 and x≈0.896

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

A≈0.037

#### 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)

By substitution u=x2∫xsinx2= -12cosx2+c

**2) Given:**

y=xsinx2, y=x4, x≥0

3) **Calculation:**

fx=xsinx2, and gx=x4

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=0.896

ii) the upper curve is y=x sinx(x2) and the lower curve is y=x4.

Therefore,

A=∫00.896xsinx2-x4dx

=-12cos(x2)-15x50.8960

=-12cos0.8962-150.8965-[-12cos02-1505

=-12cos0.8962-150.8965+12

A≈0.037

**Conclusion:**

i) The intersection points of the curves:

x=0 and x≈0.896

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

A≈0.037