Use the general slicing method to find the volume of the following solid.
The solid with a semicircular base of radius 17 whose cross sections perpendicular to the base and parallel to the diameter are squares
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Part 1
The volume of the solid is 
  
StartFraction 78608 Over 3 EndFraction
 cubic units. 
​(Type an exact​ answer.)

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.

Use the general slicing method to find the volume of the following solid. The solid with a semicircular base of radius 17 whose cross sections perpendicular to the base and parallel to the diameter are squares. The volume of the solid is StartFraction 78608 Over 3 EndFraction cubic units. (Type an exact answer.)

Learn how to find the volume of a solid with a semicircular base of radius 17, using square cross-sections perpendicular to the base. This problem involves integral calculus, using the general slicing method and symmetry in definite integrals. The exact volume of the solid is 78608/3 cubic units.

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