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We are asked to find the volume of a solid where the base is a triangular region, and cross-sections perpendicular to the \( y \)-axis are semicircles.

### Step 1: Understanding the triangular base
The triangular base has vertices at:
\[
(0, 0), (3, 0), (0, 2)
\]
This defines a right triangle in the \( xy \)-plane, with the hypotenuse forming a line segment between \( (0, 2) \) and \( (3, 0) \). The equation of the hypotenuse can be found by determining the slope:
\[
\text{slope} = \frac{0 - 2}{3 - 0} = -\frac{2}{3}
\]
Thus, the equation of the line is:
\[
y = -\frac{2}{3}x + 2
\]
Solving for \( x \) in terms of \( y \), we get:
\[
x = \frac{3}{2}(2 - y)
\]
This is the width of the triangle at any point \( y \).

### Step 2: Semicircular cross-sections
The diameter of each semicircular cross-section is the width of the triangle at a given \( y \), which is \( x = \frac{3}{2}(2 - y) \).

The area of a semicircle with diameter \( d \) is:
\[
A = \frac{1}{2} \pi \left( \frac{d}{2} \right)^2 = \frac{\pi}{8} d^2
\]
Thus, for the semicircles in this case, the area at any \( y \) is:
\[
A(y) = \frac{\pi}{8} \left( \frac{3}{2}(2 - y) \right)^2 = \frac{9\pi}{32} (2 - y)^2
\]

### Step 3: Volume by integration
The volume is found by integrating the area of the cross-sections along the \( y \)-axis, from \( y = 0 \) to \( y = 2 \):
\[
V = \int_0^2 A(y) \, dy = \int_0^2 \frac{9\pi}{32} (2 - y)^2 \, dy
\]

Let's compute this integral.

\[
V = \frac{9\pi}{32} \int_0^2 (2 - y)^2 \, dy
\]
We can expand \( (2 - y)^2 \) as:
\[
(2 - y)^2 = 4 - 4y + y^2
\]
So the integral becomes:
\[
V = \frac{9\pi}{32} \int_0^2 (4 - 4y + y^2) \, dy
\]
Now, integrating term by term:
\[
\int_0^2 4 \, dy = 4y \Big|_0^2 = 8
\]
\[
\int_0^2 -4y \, dy = -2y^2 \Big|_0^2 = -8
\]
\[
\int_0^2 y^2 \, dy = \frac{y^3}{3} \Big|_0^2 = \frac{8}{3}
\]
Adding these up:
\[
V = \frac{9\pi}{32} \left( 8 - 8 + \frac{8}{3} \right) = \frac{9\pi}{32} \times \frac{8}{3} = \frac{72\pi}{96} = \frac{3\pi}{4}
\]

Thus, the volume of the solid is:
\[
\boxed{\frac{3\pi}{4}}
\]

Would you like more details on any step?

### Related questions:
1. How would the result change if the cross-sections were full circles instead of semicircles?
2. What if the base was another geometric shape, like a rectangle, with semicircular cross-sections?
3. Can you derive the formula for the volume of solids with equilateral triangle cross-sections?
4. How would you compute the volume if the triangular base was rotated around the \( y \)-axis?
5. How do we compute the volume of solids with varying types of cross-sections (e.g., squares or rectangles)?

### Tip:
When working with solids of known cross-sections, always express the cross-section's area as a function of the variable along which you are integrating.

Use calculus to find the volume of the following solid S: The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.

This problem finds the volume of a solid where the base is a triangular region with vertices (0, 0), (3, 0), and (0, 2), and the cross-sections perpendicular to the y-axis are semicircles. The solution involves integration and geometric cross-sections.

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