Question

Consider the solid obtained by rotating the region bounded by the curves y = x - 1, y = 0, x = 6 about the x-axis. What is the shape of the solid obtained?

1) Cylinder
2) Cone
3) Sphere
4) None of the above

Answer 1

Rotating the region bounded by y = x - 1, y = 0, and x = 6 about the x-axis forms a cone, as the linear function represents a slanting line which, upon rotation, generates the conical shape.

Step-by-step explanation:

When the region bounded by the curves y = x - 1, y = 0, and x = 6 is rotated about the x-axis, the shape of the solid obtained is a cone. This is because the linear function y = x - 1 represents a slanting straight line, and when this line (together with the x-axis, which serves as the base of this region) is rotated around the x-axis, a conical shape is formed. The vertex of the cone would be at the point (1, 0), where the line intersects the x-axis, and the base would be a circle lying on the plane y = 0 with a radius equivalent to the y-value of the line at x = 6, which is 5 units.