Using a graph, it can be shown that the roots of the equation x V4 – x² = 2 – x are x x 0.81 and x = 2. Use this information to approximate the area of the region bounded by the curve y = x² V4 – x2 and the line y = 2 – x.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45e609d3-2641-46cb-87ed-a1dea6e1b55e%2Fb22aff6f-cd8e-4ece-98d6-59bca1c3f051%2Fp2dqhljp_processed.png&w=3840&q=75)
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