Animated Solution for Mathematics - Definite Integration: Find the area of the region bounded by the curves $y=x^2,y=\left|2-x^2\right|$ and $y=2$, which lies to the right of the line $x=1$.

To find the area trapped by the curves $y=x^2$, $y=|2-x^2|$, and $y=2$ for $x>1$, we must first understand the behavior of the modulus function $y=|2-x^2|$.

The function $y=|2-x^2|$ is defined piecewise: For $x^2<2$ (i.e., $1<x<\sqrt{2}$), $y=2-x^2$. For $x^2>2$ (i.e., $x>\sqrt{2}$), $y=x^2-2$.

The critical transition point occurs at $x=\sqrt{2}$.

We are restricted to the region where $x>1$. By analyzing the intersections, we identify two distinct intervals for integration:

1. Region 1 ($1<x<\sqrt{2}$): The parabola $y=x^2$ acts as the upper boundary, and the modulus curve $y=2-x^2$ acts as the lower boundary. 2. Region 2 ($\sqrt{2}<x<2$): The horizontal line $y=2$ acts as the upper boundary, and the modulus curve $y=x^2-2$ acts as the lower boundary.

The total area $A$ is the sum of the integrals over these two regions:

Simplifying the integrands, we obtain:

Now, we perform the integration for each part:

For the first integral:

For the second integral:

Summing these two results together:

The final area is $\frac{20}{3}-4\sqrt{2}$ square units.