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To solve the given problem, we need to find:

### (i) **The area of the shaded region**:
The area is bounded by the two curves:
- \( y = 8 - 2x \) (a straight line)
- \( y = x^2 \) (a parabola)

The shaded region extends between \( x = 0 \) and \( x = 4 \), as these are the intersection points of the curves.

#### Step 1: **Find the points of intersection**.
To find the points where the curves intersect, we set the equations equal:
\[
x^2 = 8 - 2x
\]
Rearranging the equation:
\[
x^2 + 2x - 8 = 0
\]
We can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 2 \), and \( c = -8 \).

#### Step 2: **Set up the integral for the area**.
The area between the curves is calculated by:
\[
\text{Area} = \int_{0}^{x_2} \left( (8 - 2x) - x^2 \right) \, dx
\]
where \( x_2 \) is the upper limit, the point of intersection.

Let me calculate the roots of the quadratic and then compute the area using integration.The points of intersection between the curves \( y = x^2 \) and \( y = 8 - 2x \) occur at \( x = -4 \) and \( x = 2 \). Since we are only concerned with the region between \( x = 0 \) and \( x = 2 \), the limits of integration are from \( 0 \) to \( 2 \).

The **area of the shaded region** is approximately:
\[
\text{Area} = 9.33 \text{ square units}.
\]

### (ii) **Volume of the solid of revolution**:
To find the volume when the shaded region is rotated about the y-axis, we use the method of cylindrical shells.

The volume formula is:
\[
V = 2\pi \int_{x_0}^{x_2} x \left( (8 - 2x) - x^2 \right) dx
\]
where \( x_0 = 0 \) and \( x_2 = 2 \).

Let me calculate the volume of the solid.The **volume of the solid** formed by rotating the shaded region about the y-axis is approximately:
\[
V = 41.89 \, \text{cubic units}.
\]

Would you like more details or have any questions about these steps?

Here are some follow-up questions related to this problem:
1. How do you find the area between two curves in general?
2. What is the method of cylindrical shells, and when is it used?
3. How would the method differ if we rotated around the x-axis instead of the y-axis?
4. What are other methods for finding the volume of a solid of revolution?
5. Can you derive the formula for volume using the disk method?

**Tip:** Always check the limits of integration carefully, especially when working with intersections between curves.

Find the area of the shaded region bounded by the curve y = x^2, the straight line y = 8 - 2x, and the x-axis.

Learn how to calculate the area of a shaded region bounded by a parabola and a straight line using definite integration, as well as finding the volume of the solid of revolution when the region is rotated around the y-axis. This step-by-step guide covers integral calculus concepts such as the method of cylindrical shells.

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