find volume of y=3-x rotatteaed around y=4 with discs

To find the volume of the solid formed by rotating the region bounded by the line \( y = 3 - x \) around the line \( y = 4 \) using the disk method, we will follow these steps:

### Step 1: Understand the Boundaries and Setup
1. The line \( y = 3 - x \) intersects the \( y \)-axis at \( (0, 3) \) and the \( x \)-axis at \( (3, 0) \).
2. We are rotating this line around the line \( y = 4 \). This rotation creates a solid of revolution.

### Step 2: Setup the Disk Method Formula
The volume \( V \) of a solid of revolution generated by rotating a function \( y = f(x) \) around a horizontal line \( y = c \) (where \( c > f(x) \)) is given by:
\[
V = \pi \int_{a}^{b} [r(x)]^2 \, dx
\]
where \( r(x) = c - f(x) \) is the distance from the function to the line of rotation.

### Step 3: Apply the Formula to the Problem
For this problem:
- The function \( f(x) = 3 - x \).
- The line of rotation is \( y = 4 \).
- The limits of integration \( a = 0 \) and \( b = 3 \).

The distance from \( y = 4 \) to \( y = 3 - x \) is:
\[
r(x) = 4 - (3 - x) = 1 + x
\]

### Step 4: Integrate to Find the Volume
Now, plug in \( r(x) = 1 + x \) into the volume formula:
\[
V = \pi \int_{0}^{3} (1 + x)^2 \, dx
\]

Expand the square and integrate:
\[
(1 + x)^2 = 1 + 2x + x^2
\]
Thus, the volume integral becomes:
\[
V = \pi \int_{0}^{3} (1 + 2x + x^2) \, dx
\]

Compute the integral:
\[
V = \pi \left[ x + x^2 + \frac{x^3}{3} \right]_{0}^{3}
\]

Evaluate the definite integral:
\[
V = \pi \left( (3 + 9 + 9) - (0 + 0 + 0) \right)
\]
\[
V = \pi (3 + 9 + 3) = 21\pi
\]

### Final Answer
The volume of the solid obtained by rotating the region bounded by \( y = 3 - x \) around the line \( y = 4 \) is \( \boxed{21\pi} \) cubic units.

Would you like more details on any step or have any questions?

Here are some related questions you might find interesting:

1. How do you set up the washer method if rotating around a different line?
2. What would the volume be if rotated around the x-axis instead of y = 4?
3. How do you handle rotations around vertical lines using the shell method?
4. What if the function was \( y = x^2 \) instead of \( y = 3 - x \)?
5. Can we use polar coordinates for any rotation problem?

**Tip:** When rotating around a line that is not one of the coordinate axes, always carefully determine the distance from the function to the axis of rotation.

Learn how to calculate the volume of a solid formed by rotating the region bounded by y = 3 - x around the line y = 4 using the disk method. Detailed step-by-step solution with integral calculus and disk method formulas.

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