Applications of integrals
An integral adds up infinitely many thin pieces. Which quantities, from an area or a volume to the work done lifting a load, can be built that way?
Where you are 10 inside
What this area is about
Every application here follows one pattern. Cut the quantity into thin pieces, approximate each piece by something simple (a rectangle, a disk, a shell, a straight segment), add the pieces, and let them shrink. The sum becomes an integral, usually evaluated by the fundamental theorem of calculus.
How these fit together
Totals from rates. Accumulation and net change, an area of its own, integrates a rate to get a total change or an amount. Work adds force times distance over small displacements.
Plane regions. Area between curves integrates top minus bottom. Average value divides \int_a^b f(x)\,dx by b - a, for f continuous on [a, b].
Volumes. Known cross sections integrates the slice area A(x). Disks and washers are the case where each slice of a solid of revolution is a circle or a ring; shells slice parallel to the axis instead.
Lengths and surfaces. Arc length adds tiny straight segments along a curve, and surface area of revolution adds the thin bands that arc sweeps out.
A path through it
Net change first, then area between curves, which average value, arc length and disks all need. Then cross sections, disks, washers and shells, since each method refers to the ones before. Arc length, then surface area, which needs both arc length and disks. Work, which borrows the slicing habit of the volume methods, comes last.
Next comes arc length and speed for parametric curves.