#### To determine

**To:**

The sketch the region enclosed by the given curves and find its area.

#### Answer

i) Region enclosed by the given curves

ii) Area of the enclosed region:19(22-1)

#### 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=xx2+1, y=x2x3+1

3) **Calculation:**

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

xx2+1=x2x3+1

xx2+1-x2x3+1=0

xx2+1-xx3+1=0

This gives,

x=0 or x2+1-xx3+1=0

x2+1=xx3+1

x2+12=xx3+12

x2+1=x5+x2

1=x5

Solving this for x gives,

x=1

Therefore, the boundary points of the region are x=0 and x=1

From the figure, we can see that the top boundary is y=xx2+1

And the bottom boundary is y=x2x3+1

A=∫01xx2+1-x2x3+1 dx

A= ∫01xx2+1 dx-∫01x2x3+1 dx

Consider ∫01xx2+1 dx

Using u substitution,

∫01xx2+1 dx

u=x2+1, du=2x dx

At x=0, u=1 and at x=1, u=2

Therefore,

∫01xx2+1 dx=∫1212u12 du

Now, consider ∫01x2x3+1 dx

Using substitution,

v=x3+1, dv=3x2 dx

At x=0, v=1 and x=1, v=2

Therefore,

∫01x2x3+1 dx= ∫1213v1/2 dv

Now the expression for area becomes

A= ∫1212u12 du-∫1213v12 dv

Integrating this gives,

A=13u3221-29v3221

Plugging the values

A=232-13-492-29

A=292-19

A=19 22-1

**Conclusion:**

i) The sketch of the region enclosed by the curves.

ii) Area of the shaded region

A=19 22-1