To find the volume of the solid using the general slicing method, let's break it down step by step:

Step 1: Understanding the Problem

We are dealing with a solid with:

• A semicircular base of radius $r = 17$.
• Square cross-sections perpendicular to the base and parallel to the diameter.

The goal is to compute the volume of this solid by integrating the area of the cross-sections.

Step 2: Equation of the Semicircle

The base of the solid is a semicircle, and we can write its equation (in terms of $x$ and $y$) as: $y = \sqrt{r^2 - x^2}$ where $r = 17$. So the equation of the semicircle becomes: $y = \sqrt{17^2 - x^2} = \sqrt{289 - x^2}$ This semicircle is on the interval $-17 \leq x \leq 17$ along the x-axis.

Step 3: Area of the Cross-Sections

The cross-sections perpendicular to the base and parallel to the diameter are squares. The side length of each square is the length of the vertical line from $y = 0$ to the top of the semicircle at any given $x$. This length is $2y$, where: $y = \sqrt{289 - x^2}$ Thus, the side length of each square is: $s(x) = 2y = 2\sqrt{289 - x^2}$ The area of each square cross-section is: $A(x) = s(x)^2 = \left( 2\sqrt{289 - x^2} \right)^2 = 4(289 - x^2)$

Step 4: Volume of the Solid

To find the volume, we integrate the area of the cross-sections along the interval $-17 \leq x \leq 17$. The volume is given by: $V = \int_{-17}^{17} A(x) \, dx = \int_{-17}^{17} 4(289 - x^2) \, dx$

Step 5: Evaluate the Integral

Now, let's compute the integral: $V = 4 \int_{-17}^{17} (289 - x^2) \, dx$ Since $289 - x^2$ is an even function, we can simplify the integral by doubling the integral over the interval $[0, 17]$: $V = 8 \int_0^{17} (289 - x^2) \, dx$ Now, integrate term by term: $\int_0^{17} 289 \, dx = 289x \bigg|_0^{17} = 289(17 - 0) = 4913$ $\int_0^{17} x^2 \, dx = \frac{x^3}{3} \bigg|_0^{17} = \frac{17^3}{3} = \frac{4913}{3}$ So, the volume is: $V = 8 \left( 4913 - \frac{4913}{3} \right) = 8 \left( \frac{14739}{3} - \frac{4913}{3} \right) = 8 \times \frac{9836}{3} = \frac{78608}{3}$

Final Answer:

The volume of the solid is: $V = \frac{78608}{3} \text{ cubic units.}$

Would you like further clarification or any additional details?

Here are 5 related questions you might find helpful:

1. How do you apply the slicing method to solids with triangular cross-sections?
2. What is the general formula for the volume of a solid with square cross-sections?
3. How would the volume change if the base were a full circle instead of a semicircle?
4. Can the slicing method be used to find volumes of solids with irregular cross-sections?
5. How does the radius of the semicircular base affect the final volume?

Tip: When dealing with symmetry in volume integrals, exploit it to simplify calculations by integrating over half or a quarter of the domain.