The problem provided involves calculating the volume of an object where semicircular cross-sections are perpendicular to the base, which is outlined in the $xy$-plane. To compute the volume of this object, we need to follow these steps:

Step-by-step solution:

1. Identify the base of the object: The base is the blue shape in the $xy$-plane, as shown on the left side of the image. It spans from $x = -1$ to $x = 1$ along the $x$-axis, and the height of the shape reaches from $y = 0$ to $y = 6$.

2. Find the equation of the shape:

   • From $x = -1$ to $x = 1$, the shape resembles a trapezoid with vertical boundaries.
   • For the height $h(y)$ at each $y$-value, the width at the bottom is 0 at $y = 0$, and it reaches a maximum of 2 units wide (from $x = -1$ to $x = 1$) at $y = 6$.

   The width of the base at a given height $y$ is proportional to $y$. The function describing the width is: w(y) = \frac{y}{3} \quad \text{(since the width increases from 0 to 2 as y goes from 0 to 6)}.

3. Volume formula using semicircular cross-sections: For each semicircular cross-section, the diameter is the width $w(y)$. The radius $r(y)$ is half of this width: $r(y) = \frac{w(y)}{2} = \frac{y}{6}.$ The area $A(y)$ of a semicircle with radius $r(y)$ is given by: $A(y) = \frac{1}{2} \pi r(y)^2 = \frac{1}{2} \pi \left( \frac{y}{6} \right)^2 = \frac{\pi}{72} y^2.$

4. Integrate to find the volume: The volume of the solid is found by integrating the area of the cross-sections along the $y$-axis, from $y = 0$ to $y = 6$: $V = \int_0^6 A(y) \, dy = \int_0^6 \frac{\pi}{72} y^2 \, dy.$

5. Solve the integral: Now we compute the integral: $V = \frac{\pi}{72} \int_0^6 y^2 \, dy = \frac{\pi}{72} \left[ \frac{y^3}{3} \right]_0^6 = \frac{\pi}{72} \cdot \frac{6^3}{3} = \frac{\pi}{72} \cdot \frac{216}{3} = \frac{\pi}{72} \cdot 72 = \pi.$

Thus, the exact value of the volume is:

$V = \pi.$

Would you like further details or have any questions?

Related questions:

1. How do we derive the formula for the area of a semicircle?
2. What would the volume be if the cross-sections were full circles instead of semicircles?
3. Can this method be applied if the cross-sections are equilateral triangles instead of semicircles?
4. How does the width function $w(y)$ change if the shape is a different polygon?
5. What would happen to the volume if the shape were rotated around the $y$-axis?

Tip:

When dealing with solids of known cross-sections, always start by finding an expression for the area of the cross-section in terms of the axis you're integrating along.