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 find the volume of the solid formed by stacking these rectangular cross-sections from \(x=0\) to \(x=3\), we need to integrate the area of the rectangles. The width of the rectangle is \(x\), and the height is given by \(3 - x\). Therefore, the area \(A\) of each rectangle as a function of \(x\) is: \[ A(x) = \text{width} \times \text{height} = x(3 - x) = 3x - x^2. \] Next, we need to calculate the volume \(V\) of the solid by integrating the area from \(0\) to \(3\): \[ V = \int_{0}^{3} A(x) \, dx = \int_{0}^{3} (3x - x^2) \, dx. \] Computing the integral: \[ V = \int_{0}^{3} (3x - x^2) \, dx = \left[\frac{3x^2}{2} - \frac{x^3}{3}\right]_{0}^{3}. \] Evaluating the integral at the bounds: \[ V = \left(\frac{3(3)^2}{2} - \frac{(3)^3}{3}\right) - \left(\frac{3(0)^2}{2} - \frac{(0)^3}{3}\right). \] Calculating the terms: \[ = \left(\frac{27}{2} - 9\right) - 0 = \frac{27}{2} - \frac{18}{2} = \frac{9}{2}. \] Thus, the volume of the solid is: \[ \boxed{\frac{9}{2}}. \]
