Final answer:
To sketch the region, draw the line y=4x, the x-axis, and the vertical line at x=2 to form a triangle. Revolving this region around the x-axis creates a cone. Using the formula for the volume of a cone, you find the volume to be (128/3)π cubic units.
Step-by-step explanation:
To sketch the region bounded by the curves y=4x, y=0, and x=2, you start by drawing the line y=4x, which is a straight line through the origin with a slope of 4, progressing upwards to the right. The line y=0 is simply the x-axis, and the vertical line at x=2 is drawn perpendicular to the x-axis at the point x = 2. The region bounded by these three is a right triangle with height 8 units (since when x=2, y=4x=8) and base of 2 units along the x-axis.
When you revolve this region about the x-axis, it forms a cone. To find the volume of the solid cone, you can use the formula for the volume of a cone which is one-third of the base area times the height, V = (1/3)πr²h, where the base of the solid is a circle with radius equal to the height of the triangle (8 units).
Therefore, the volume V is V = (1/3)π(8²)(2), which simplifies to V = (1/3)π(64)(2) or V = (128/3)π cubic units.