Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = 43y
,   x = 0,   y = 3;
   about the y-axis

To find the volume \( V \) of the solid obtained by rotating the region bounded by the given curves about the \( y \)-axis, we'll use the method of disks or washers. Given curves are:

- \( x = 43y \)
- \( x = 0 \) (the y-axis)
- \( y = 3 \)

### Step-by-Step Solution

1. **Identify the Bounded Region:**
   The region is bounded by:
   - The line \( x = 43y \), which represents a linear function.
   - The y-axis, \( x = 0 \).
   - The horizontal line \( y = 3 \).

2. **Determine the Limits of Integration:**
   Since we are rotating about the \( y \)-axis, we will integrate with respect to \( y \). The limits for \( y \) are from 0 to 3.

3. **Express \( x \) as a Function of \( y \):**
   The given equation is already in terms of \( y \):
   \[
   x = 43y
   \]

   Since we are rotating around the \( y \)-axis, the radius of the disk at any point \( y \) is given by \( x = 43y \).

4. **Set Up the Integral for the Volume:**
   The formula for the volume of a solid of revolution around the \( y \)-axis using the disk method is:
   \[
   V = \pi \int_{a}^{b} [R(y)]^2 \, dy
   \]
   where \( R(y) = x = 43y \) and the limits of integration are \( y = 0 \) to \( y = 3 \).

   Substituting the radius \( R(y) = 43y \):
   \[
   V = \pi \int_{0}^{3} (43y)^2 \, dy
   \]

5. **Compute the Integral:**
   First, square the radius:
   \[
   (43y)^2 = 1849y^2
   \]

   So, the integral becomes:
   \[
   V = \pi \int_{0}^{3} 1849y^2 \, dy
   \]

   Now integrate with respect to \( y \):
   \[
   V = \pi \left[ \frac{1849y^3}{3} \right]_{0}^{3}
   \]

   Evaluate the definite integral:
   \[
   V = \pi \left( \frac{1849 \cdot 3^3}{3} - \frac{1849 \cdot 0^3}{3} \right)
   \]

   Simplify further:
   \[
   V = \pi \left( \frac{1849 \cdot 27}{3} \right)
   \]

   \[
   V = \pi \cdot 1849 \cdot 9
   \]

   \[
   V = 16641\pi
   \]

### Final Answer:

The volume \( V \) of the solid obtained by rotating the region about the \( y \)-axis is:

\[
V = 16641\pi \, \text{cubic units}
\]

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### Related Questions:

1. How would the volume change if we rotated the region around the line \( y = 4 \)?
2. What is the volume if the region is rotated about the line \( x = -2 \)?
3. How do we find the surface area of the solid instead of the volume?
4. How does changing the function from linear to quadratic affect the volume calculation?
5. What if the function \( x = 43y \) is replaced with \( x = 43y^2 \)?

#### Tip:
When rotating a region about the \( y \)-axis, ensure that your integral is set up in terms of \( y \) with the correct limits and radius function!

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 43y, x = 0, y = 3; about the y-axis.

Learn how to calculate the volume of a solid formed by rotating the region bounded by x = 43y, x = 0, and y = 3 about the y-axis using the disk method. Includes step-by-step integration for students in Grades 11-12.

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