Find the volume of the solid of revolution of the region ℛ enclosed by the parabola

𝑥2 = 4 − 𝑦 and the straight line 𝑥 + 𝑦 = 2 about the 𝑥-axis.

To find the volume of the solid of revolution of the region \( \mathcal{R} \), enclosed by the parabola \( x^2 = 4 - y \) and the straight line \( x + y = 2 \), about the \( x \)-axis, we can use the **method of cylindrical shells** or the **disk/washer method**. Let's use the **disk/washer method** since we're rotating around the \( x \)-axis.

### Step 1: Find the points of intersection
We first find the points where the parabola \( x^2 = 4 - y \) and the straight line \( x + y = 2 \) intersect. 

From the straight line equation, solve for \( y \):
\[
y = 2 - x.
\]
Substitute this into the equation of the parabola:
\[
x^2 = 4 - (2 - x),
\]
\[
x^2 = 4 - 2 + x,
\]
\[
x^2 - x - 2 = 0.
\]
Solve this quadratic equation for \( x \).

### Step 2: Set up the volume integral
Once we find the intersection points, we'll express the region bounded by the curves in terms of the vertical distance from the x-axis (i.e., \( y \)). The volume of the solid generated by revolving the region around the \( x \)-axis is given by the formula:

\[
V = \pi \int_{a}^{b} \left[ R_{\text{outer}}^2 - R_{\text{inner}}^2 \right] dy,
\]
where \( R_{\text{outer}}(y) \) is the distance from the \( x \)-axis to the outer curve, and \( R_{\text{inner}}(y) \) is the distance from the \( x \)-axis to the inner curve. 

Let me first solve for the points of intersection, then proceed with the integral setup.# Solve the quadratic equation x^2 - x - 2 = 0
x = sp.Symbol('x')
quadratic_eq_intersection = x**2 - x - 2

# Solve the equation for x to find the points of intersection
x_solutions = sp.solve(quadratic_eq_intersection, x)
x_solutionsThe points of intersection are \( x = -1 \) and \( x = 2 \).

### Step 3: Set up the volume integral
To set up the volume integral using the washer method, we revolve the region around the \( x \)-axis. We need to express the curves in terms of \( y \).

1. **For the parabola:** The equation is \( x^2 = 4 - y \), so
   \[
   x = \pm \sqrt{4 - y}.
   \]
   Taking the positive branch (since we're considering the right side of the parabola), we get:
   \[
   x_{\text{parabola}}(y) = \sqrt{4 - y}.
   \]

2. **For the straight line:** The equation is \( x + y = 2 \), so
   \[
   x = 2 - y.
   \]

We integrate from \( y = 0 \) to \( y = 3 \) (as found from the points of intersection). The volume is given by:
\[
V = \pi \int_{0}^{3} \left( (\sqrt{4 - y})^2 - (2 - y)^2 \right) dy.
\]
Let's compute the volume now.# Define the variables and the expressions for the volume integral
y = sp.Symbol('y')

# Define the outer radius (parabola) and inner radius (straight line)
R_outer = sp.sqrt(4 - y)
R_inner = 2 - y

# Set up the volume integral using the washer method
volume_integral = sp.pi * sp.integrate(R_outer**2 - R_inner**2, (y, 0, 3))
volume_integralThe volume of the solid of revolution of the region \( \mathcal{R} \), enclosed by the parabola \( x^2 = 4 - y \) and the straight line \( x + y = 2 \), about the \( x \)-axis is:

\[
V = \frac{9\pi}{2}.
\]

Would you like to explore this further or ask any questions?

Here are five related questions:
1. How does the washer method differ from the cylindrical shells method?
2. Can we find the surface area of this solid of revolution?
3. What happens if we rotate the region about the \( y \)-axis instead of the \( x \)-axis?
4. How does the intersection of curves help in defining the limits of integration?
5. What are some real-world applications of solids of revolution?

**Tip:** For complex regions, ensure the correct identification of inner and outer radii when setting up the washer method.

Learn how to find the volume of the solid of revolution generated by the intersection of the parabola x^2 = 4 - y and the line x + y = 2 around the x-axis. Detailed solution using the disk/washer method and integral calculus.

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