In the disc method, the radius R represents:

In the disc method, the radius R represents:

• A. the distance from the axis of rotation to the top function
• B. the area of the cross section
• C. the height of the cross section
• D. the average value of f(x) over [a,b]

Answer: (A)

In the disc method, each cross-section perpendicular to the axis of rotation is a disk whose size is determined by how far the region is from that axis. The radius is this distance to the outer boundary of the region, i.e., to the graph of the function. So when you rotate the area under y = f(x) around the x-axis, the radius at position x is R = f(x). The cross-sectional area is πR^2, and integrating that along the interval gives the volume. This radius is a distance, not an area or a height, and it changes with x just like the function value does. If you rotate around a different axis, the radius would be the distance from that axis to the function, i.e., |f(x) − axis|.

In the disc method, each cross-section perpendicular to the axis of rotation is a disk whose size is determined by how far the region is from that axis. The radius is this distance to the outer boundary of the region, i.e., to the graph of the function. So when you rotate the area under y = f(x) around the x-axis, the radius at position x is R = f(x). The cross-sectional area is πR^2, and integrating that along the interval gives the volume. This radius is a distance, not an area or a height, and it changes with x just like the function value does. If you rotate around a different axis, the radius would be the distance from that axis to the function, i.e., |f(x) − axis|.