   Chapter 5.R, Problem 6E

Chapter
Section
Textbook Problem

# Find the area of the region bounded by the given curves. y = x ,    y = x 2 ,    x = 2

To determine

To find:

The area of the region bounded by the given curves.

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 fxg(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+

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