Find the total area of the shaded regions

To find the total area of the shaded regions in the diagram, we need to use principles from coordinate geometry and calculus, specifically involving the areas under curves and between curves. Let's break down the problem step by step.

Problem Breakdown:

Identify the curves and lines involved in the diagram.

Determine the points of intersection.

Set up the integrals to find the areas of the regions.

Compute the total area.

Step-by-Step Solution:

Given Information:

We have the curves y=xy=x

and y=x2y=x2.

We need to find the area of the regions shaded between these curves.

1. Identify the curves and their intersection points:

The given curves are:

y=xy=x

y=x2y=x2

To find the points of intersection, we set:

x=x2x

=x2

Squaring both sides, we get:

x=x4x=x4

Solving for xx, we factorize:

x(1−x3)=0x(1−x3)=0

This gives us:

x=0orx3=1x=0orx3=1

x=0orx=1x=0orx=1

Thus, the points of intersection are at x=0x=0 and x=1x=1.

2. Determine the regions:

The region between the curves from x=0x=0 to x=1x=1 can be found by integrating the difference between the top function and the bottom function over this interval. From x=0x=0 to x=1x=1:

y=xy=x

is above y=x2y=x2.

3. Set up the integral:

The area AA of the shaded region between the curves from x=0x=0 to x=1x=1 is given by:

A=∫01(x−x2) dxA=∫01(x

−x2)dx

4. Compute the integral:

Evaluate the integral:

A=∫01x dx−∫01x2 dxA=∫01x

dx−∫01x2dx

Each integral can be computed separately:

Compute ∫01x dx∫01x

dx:

Rewrite xx

as x1/2x1/2:

∫01x1/2 dx=[23x3/2]01=23(13/2−03/2)=23∫01x1/2dx=[32x3/2]01=32(13/2−03/2)=32

Compute ∫01x2 dx∫01x2dx:

∫01x2 dx=[13x3]01=13(13−03)=13∫01x2dx=[31x3]01=31(13−03)=31

Subtract the two integrals:

A=23−13=13A=32−31=31

Final Answer:

The total area of the shaded region is:

1331