Volumes by Cross-Sections — Calc II Explorer

Slices perpendicular to

the x-axis the y-axis

Base region

These are the curves the slices run between. One curve on its own is read against the axis you are slicing perpendicular to — y = 0 for the x-axis, x = 0 for the y-axis — so a second equation is only needed when the region lies between two curves. The independent variable is read off each equation (y = sqrt(x), x = y^2), so rows may mix the two forms, and how far the slices run is the interval below rather than an equation.

Cross-section

Square Circle Semicircle Right △

Each slice of the region is a chord of length s. The shape stands on that chord, perpendicular to the base: a square of side s, or a right triangle whose base is the chord, with the right angle at one end and the acute angle θ at the other, so its height is s·tan θ. A circle takes the chord as its diameter, so it reaches as far below the base as above. A semicircle takes the chord as one of its radii, standing on the axis the chord starts from and doming up through the region — for a region resting on the axis that is the top half of the solid the same region would sweep out by revolving.

Interval (x)

a b

auto-detect from intersections

Base region & chord

slice

Base region & chord

Build-up

Solid + cross-section