(c) Region #3: Boundcd by yi = x2 + 2, y2 = 4 - a?, and y3 = 2 - x. %3D 3.

Calculus: Early Transcendentals
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Set up the Area integrals for the regions bounded by the given curves. Show work for finding the exact intersection points. Do not integrate or evaluate the integral. Area Integral: set up)

### Calculating the Bounded Area

**Region #3: Definition**

This region is bounded by the following curves:
- \( y_1 = -x^2 + 2 \)
- \( y_2 = 4 - x^2 \)
- \( y_3 = 2 - x \)

**Diagram Explanation**

The graph provided displays three curves in the Cartesian plane:

1. **Curve \( y_1 = -x^2 + 2 \):** 
   - This is a downward-opening parabola. It crosses the y-axis at \( y = 2 \).

2. **Curve \( y_2 = 4 - x^2 \):**
   - This is another downward-opening parabola, wider than \( y_1 \), crossing the y-axis at \( y = 4 \).

3. **Curve \( y_3 = 2 - x \):**
   - This is a straight line with a negative slope. It intersects the y-axis at \( y = 2 \) and the x-axis at \( x = 2 \).

The shaded region indicates the area enclosed by these curves.

**Area Integral Setup**

To find the area of the bounded region, set up the integral with respect to \( x \). The integral expression should be organized by identifying where each function is the upper and lower boundary in the overlapping region:

\[ 
\text{Area Integral:} \quad \int_{x_\text{left}}^{x_\text{right}} (\text{upper function} - \text{lower function}) \, dx 
\]

Make sure to determine \( x_\text{left} \) and \( x_\text{right} \) by finding the points of intersection between the curves.
Transcribed Image Text:### Calculating the Bounded Area **Region #3: Definition** This region is bounded by the following curves: - \( y_1 = -x^2 + 2 \) - \( y_2 = 4 - x^2 \) - \( y_3 = 2 - x \) **Diagram Explanation** The graph provided displays three curves in the Cartesian plane: 1. **Curve \( y_1 = -x^2 + 2 \):** - This is a downward-opening parabola. It crosses the y-axis at \( y = 2 \). 2. **Curve \( y_2 = 4 - x^2 \):** - This is another downward-opening parabola, wider than \( y_1 \), crossing the y-axis at \( y = 4 \). 3. **Curve \( y_3 = 2 - x \):** - This is a straight line with a negative slope. It intersects the y-axis at \( y = 2 \) and the x-axis at \( x = 2 \). The shaded region indicates the area enclosed by these curves. **Area Integral Setup** To find the area of the bounded region, set up the integral with respect to \( x \). The integral expression should be organized by identifying where each function is the upper and lower boundary in the overlapping region: \[ \text{Area Integral:} \quad \int_{x_\text{left}}^{x_\text{right}} (\text{upper function} - \text{lower function}) \, dx \] Make sure to determine \( x_\text{left} \) and \( x_\text{right} \) by finding the points of intersection between the curves.
Expert Solution
Step 1

To set the integral to find the area bounded by the curves:

y1=x2+2, y2=4-x2 & y3=2-x

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