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 \(x=3\).

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