A solid lies between planes perpendicular to the​ x-axis at xequals0 and xequals12. The​ cross-sections perpendicular to the axis on the interval 0less than or equalsxless than or equals12 are squares with diagonals that run from the parabola y equals negative 2 StartRoot x EndRoot to the parabola y equals 2 StartRoot x EndRoot. Find the volume of the solid.

We are tasked with finding the volume of a solid whose cross-sections perpendicular to the \(x\)-axis are squares. The diagonal of each square runs from the parabola \(y = -2\sqrt{x}\) to \(y = 2\sqrt{x}\), for \(x\) ranging from \(0\) to \(12\).

### Step 1: Determine the length of the diagonal of each square
The length of the diagonal at a given \(x\)-value is the vertical distance between the two parabolas. This is calculated as:

\[
\text{Diagonal} = 2\sqrt{x} - (-2\sqrt{x}) = 4\sqrt{x}
\]

### Step 2: Relating the diagonal to the side length of the square
Let \(s(x)\) represent the side length of the square at a given \(x\)-value. Since the diagonal \(d(x)\) and the side length of a square are related by the formula \(d = s\sqrt{2}\), we can express \(s(x)\) as:

\[
s(x) = \frac{d(x)}{\sqrt{2}} = \frac{4\sqrt{x}}{\sqrt{2}} = \frac{4}{\sqrt{2}}\sqrt{x} = 2\sqrt{2}\sqrt{x}
\]

### Step 3: Calculate the area of the square cross-section
The area \(A(x)\) of each square cross-section is given by the square of the side length:

\[
A(x) = s(x)^2 = \left(2\sqrt{2}\sqrt{x}\right)^2 = 8x
\]

### Step 4: Set up the volume integral
The volume of the solid is found by integrating the area of the cross-sections from \(x = 0\) to \(x = 12\):

\[
V = \int_0^{12} A(x) \, dx = \int_0^{12} 8x \, dx
\]

### Step 5: Evaluate the integral
Now, we can evaluate the integral:

\[
V = 8 \int_0^{12} x \, dx = 8 \left[\frac{x^2}{2}\right]_0^{12} = 8 \left[\frac{12^2}{2} - \frac{0^2}{2}\right]
\]
\[
V = 8 \times \frac{144}{2} = 8 \times 72 = 576
\]

### Final Answer:
The volume of the solid is \(576\) cubic units.

---

Would you like further details or have any questions? Here are some related problems you can explore:

1. What if the cross-sections were equilateral triangles instead of squares?
2. How does the volume change if the parabolas are shifted vertically?
3. What is the volume if the diagonal of the square is instead proportional to \(x^2\)?
4. What would the integral look like if the cross-sections were semicircles?
5. How do we find the centroid of the solid in this case?

**Tip:** For volume problems involving cross-sectional areas, always start by relating the geometric property (side, diagonal, etc.) to the variable \(x\).

A solid lies between planes perpendicular to the​ x-axis at x=0 and x=12. The​ cross-sections perpendicular to the axis on the interval 0≤x≤12 are squares with diagonals that run from the parabola y = -2√x to the parabola y = 2√x. Find the volume of the solid.

Calculate the volume of a solid between x=0 and x=12 where the cross-sections perpendicular to the x-axis are squares with diagonals running between the parabolas y=-2√x and y=2√x. Learn step-by-step how to set up the integral and find the volume using key concepts from integral calculus and geometry.

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