**Transcription for Educational Website:**Using a graph, it can be shown that the roots of the equation \( x^2 \sqrt{4 - x^2} = 2 - x \) are \( x \approx 0.81 \) and \( x = 2 \). Use this information to approximate the area of the region bounded by the curve \( y = x^2 \sqrt{4 - x^2} \) and the line \( y = 2 - x \).\[ \text{Area} = \] **Explanation:**This problem involves finding the area of the region enclosed by a curve and a straight line. The curve is defined by the equation \( y = x^2 \sqrt{4 - x^2} \), which is a combination of a polynomial and a square root function, indicating the curve may have a unique shape influenced by both algebraic and transcendental aspects. The line \( y = 2 - x \) represents a straight line with a negative slope.To solve this:1. **Determine the Intersection Points:** The roots \( x \approx 0.81 \) and \( x = 2 \) are the x-values where the curve and the line intersect. These points are critical as they define the limits for the area calculation.2. **Set Up the Integral for Area:** The area between two curves from \( x = a \) to \( x = b \) is given by integrating the difference between the two functions: \[ \text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx \] In this problem, \( f(x) = 2 - x \) and \( g(x) = x^2 \sqrt{4 - x^2} \). Thus, the integral becomes: \[ \text{Area} = \int_{0.81}^{2} [(2 - x) - (x^2 \sqrt{4 - x^2})] \, dx \]3. **Evaluate the Integral:** Solve the integral to find the approximate area.Thus, with the provided roots, you can perform an integration to find the area enclosed by the specified functions.

Name: **Transcription for Educational Website:**Using a graph, it can be sh
Uploaded: 2024-03-20T06:46:14.000Z
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Description: **Transcription for Educational Website:**Using a graph, it can be shown that the roots of the equation \( x^2 \sqrt{4 - x^2} = 2 - x \) are \( x \approx 0.81 \) and \( x = 2 \). Use this information to approximate the area of the region bounded by