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.