I'll assume that the "shaded area" is defined by the x- and y-axes and the line 5x+4y=40 (and therefore is in Quadrant I only). It's triangular. One of the legs rests upon the x-axis, and x ranges from 0 to 8. The other leg rests upon the y-axis, and y ranges from 0 to 10.
Let's use the disk method to determine the volume of the solid generated if this area is revolved around the x-axis. Since 5x+4y=40, 4y=40-5x, and y = 10-(5/4)x. Imagine "slices" of this solid that are perpendicular to the x-axis and that have the radius y = r = y = 10-(5/4)x.
Then the area of any such slice is pi*r^2, or pi*(10-5x/4)^2. The thickness of each slice is dx. Thus, the volume of each slice is dv = pi*(10-5x/4)^2 dx.
Integrate this from x=0 to x=8 to determine the total volume of the solid. The result of integration is V = (266 2/3) pi, or 800pi/3. This is the vol. of the solid.
