#### To determine

**To find:**

The area of the region bounded by the given curves.

#### Answer

10-423

#### Explanation

**1) Concept:**

The area A of the region bounded by the curves y=fx, y=gx, and the lines x=a, x=b, where f and g are continuous and fx≥g(x) for all x in a, b, is

A= ∫abfx-gxdx

**2) Given:**

The region bounded by curves y=x, y=x2, x=2

**3) Calculation:**

The given region is bounded by curves y=x, y=x2, x=2. We plot the graph below

From the graph, the region is divided into two parts.

From x=0 to x=1, the upper boundary curve is y=x and the lower boundary curve is y=x2

From x=1 to x=2, the upper boundary curve is y=x2 and the lower boundary curve is y=x

Therefore, the area of the region bounded by the curves y=x, y=x2, x=2 between x=0 to x=2 is given by

A= ∫01x-x2dx+∫12x2- xdx

= ∫01x12-x2dx+∫12x2-x12dx

By using fundamental theorem of calculus and power rule,

A=x3232-x3301+x33-x323212

=2x323-x3301+x33- 2x32312

=2(1)323-(1)33-2(0)323-(0)33+(2)33-2(2)323-(1)33-2(1)323

=23-13+83-2223-13-23

=13+83-423+13

=103-423

=10-423

**Conclusion:**

The area of the region bounded by the curves y=x, y=x2, x=2 between x=0 to x=2 is A=10-423