A solid has a cross section perpendicular to the x-axis, which is a square with side length $$y$$. If the function $$y = 2x + 1$$ represents the upper boundary of the solid from $$x=0$$ to $$x=3$$, find its volume.

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To find the volume of the solid with square cross-sections perpendicular to the $$ x $$-axis, we'll follow these steps:

1. **Understand the Cross-Section:**
   - Each cross-section perpendicular to the $$ x $$-axis is a square.
   - The side length of each square is given by $$ y = 2x + 1 $$.
   - Therefore, the area $$ A(x) $$ of each square cross-section is:
     $$
     A(x) = y^2 = (2x + 1)^2
     $$

2. **Set Up the Integral for Volume:**
   - The volume $$ V $$ of the solid is obtained by integrating the area of the cross-sections from $$ x = 0 $$ to $$ x = 3 $$:
     $$
     V = \int_{0}^{3} (2x + 1)^2 \, dx
     $$

3. **Expand and Integrate:**
   - First, expand $$ (2x + 1)^2 $$:
     $$
     (2x + 1)^2 = 4x^2 + 4x + 1
     $$
   - Now, integrate term by term:
     $$
     V = \int_{0}^{3} (4x^2 + 4x + 1) \, dx = \left[ \frac{4}{3}x^3 + 2x^2 + x \right]_{0}^{3}
     $$

4. **Evaluate the Integral:**
   - Plug in the upper limit $$ x = 3 $$:
     $$
     \frac{4}{3}(3)^3 + 2(3)^2 + 3 = \frac{4}{3}(27) + 18 + 3 = 36 + 18 + 3 = 57
     $$
   - The lower limit $$ x = 0 $$ gives 0.

5. **Final Volume:**
   $$
   V = 57 \text{ cubic units}
   $$

**Answer:**  
The volume of the solid is 57 cubic units.
The volume of the solid is 57 cubic units.

A solid has a cross section perpendicular to the x-axis, which is a square with side length . If the function represents the upper boundary of the solid from to , find its volume.

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