The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = $$x$$ and height = $$3 - x$$ for $$0 \leq x \leq 3$$. Calculate the volume of the solid when these rectangles are stacked from $$x=0$$ to $$x=3$$.

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To calculate the volume of the solid formed by stacking the rectangular cross-sections from $$ x = 0 $$ to $$ x = 3 $$, we can follow these steps:

1. **Determine the Area Function:**
   The area $$ A(x) $$ of each rectangular cross-section is given by:
   $$
   A(x) = \text{width} \times \text{height} = x \times (3 - x) = 3x - x^2
   $$

2. **Set Up the Integral:**
   The volume $$ V $$ of the solid is the integral of the area function from $$ x = 0 $$ to $$ x = 3 $$:
   $$
   V = \int_{0}^{3} (3x - x^2) \, dx
   $$

3. **Compute the Integral:**
   $$
   \int (3x - x^2) \, dx = \frac{3}{2}x^2 - \frac{1}{3}x^3 + C
   $$
   Evaluating from $$ 0 $$ to $$ 3 $$:
   $$
   V = \left[ \frac{3}{2}(3)^2 - \frac{1}{3}(3)^3 \right] - \left[ \frac{3}{2}(0)^2 - \frac{1}{3}(0)^3 \right]
   $$
   $$
   V = \left[ \frac{3}{2} \times 9 - \frac{1}{3} \times 27 \right] - 0
   $$
   $$
   V = \left[ \frac{27}{2} - 9 \right]
   $$
   $$
   V = \frac{27}{2} - \frac{18}{2} = \frac{9}{2}
   $$

4. **Final Answer:**
   The volume of the solid is $$ \frac{9}{2} $$ cubic units.

**Answer:**  
The solid has volume equal to nine-halves cubic units
The volume of the solid is $$ \frac{9}{2} $$ cubic units.

The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = and height = for . Calculate the volume of the solid when these rectangles are stacked from to .

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