5. rotate about x - axis: y v = volume = type your answer... cubic units

Explanation:

Step1: Determine the shape and dimensions

The shaded - region appears to be a rectangle. From the grid, assume the length of the rectangle along the y - axis is (h = 4) units and the length along the x - axis is (w=4) units. When rotated about the x - axis, we use the disk method for finding the volume of the solid of revolution. The formula for the volume (V) of a solid of revolution of a function (y = f(x)) about the x - axis from (x=a) to (x = b) is (V=\pi\int_{a}^{b}[f(x)]^{2}dx). For a rectangle with height (y = h) and width along the x - axis, if we consider the rectangle with (y) constant from (x_1) to (x_2), the volume of the solid of revolution about the x - axis is (V=\pi r^{2}l), where (r) is the radius (height of the rectangle) and (l) is the length (width of the rectangle).

Step2: Apply the volume formula

Here, (r = 4) (height of the rectangle) and (l = 4) (width of the rectangle). Using the formula (V=\pi r^{2}l), we substitute (r = 4) and (l = 4). So (V=\pi\times4^{2}\times4).

Step3: Calculate the volume

[ \begin{align*} V&=\pi\times16\times 4\ &=64\pi \end{align*} ]

Answer:

(64\pi)