1.8.4—Area and volume

Area between curves is ∫(upper−lower)dx or ∫(right−left)dy. Rotating a region about an axis uses washers or shells, with radius and thickness tied to the chosen variable.

Find intersections first, split where the order changes, and keep units squared or cubed. A volume formula must match the axis and region.

Rotating y=x from x=0 to 2 about the x-axis gives V=π∫₀²x²dx=8π/3.

Integrating one curve without subtracting the other can count the wrong region, and a graph crossing may require separate intervals.