Shell Method with Shifted Axes
Apply shell integration after translating the coordinate axis to handle solids of revolution about non-origin x- or y-lines.
9/9 units complete · 62 records
Choosing Shell Orientation and Variable
Decide whether shifted-axis rotation should be modeled with vertical or horizontal shells and whether the integral should use dx or dy.
Shell Method About Shifted Vertical Axes
Set up shell integrals for rotation about lines of the form x = h, with radius measured as horizontal distance to the shifted axis.
Shell Method About Shifted Horizontal Axes
Set up shell integrals for rotation about lines of the form y = k, with radius measured as vertical distance to the shifted axis.
Interval Translation and Shifted Bounds
Recenter shell problems by substituting coordinates relative to the shifted axis and updating the integration limits accordingly.
Axis Crossings and Split-Interval Shells
Handle cases where the axis lies inside the interval by splitting the integral so shell radii remain nonnegative.