#### To determine

**To:**

The area of the region enclosed by the given curves.

#### Answer

i) The region enclosed by the given curves

ii) Area: 25-4

#### 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)∫xa+x2dx=a+x2+C∫xa-x2dx=-a-x2+C

**2) Given:**

y=x1+x2, y=x9-x2, x≥0

3) **Calculation:**

First, find the intersection points of the curves by solving their equations simultaneously.

x1+x2= x9-x2

This gives,

x=0 or

1+x2= 9-x2

1+x2=9-x2

Combining like terms,

2x2=8

x2=4

x=± 2

So points of intersection of the curves are at x=0, x=2, x=-2

But from the given condition, x≥0

Therefore, the boundary points of the region are

x=0 and x=2

From the figure, we can see that the top boundary is

y=x1+x2

And the bottom boundary is

y=x9-x2

So enclosed area is

A=∫02x1+x2-x9-x2 dx

Using formula

A= 1+x2+9-x220

Plugging the values

=1+22+ 9-22-1+02+ 9-02

=5+5-1+3

=25-4

**Conclusion:**

i) The region enclosed by the curves

ii) The area of the shaded region

A=25-4