area between curves

Finding the area between two curves is the first place integration stops being an abstract exercise and starts measuring real regions. Picture the sliver of land between a winding river and a road that both run across a map: to find that area you imagine slicing it into countless thin vertical strips, each strip's height being the gap between the top boundary and the bottom boundary at that spot.

Each thin strip has area (top - bottom) times its tiny width dx, and integrating adds them all up: area = integral from a to b of (top(x) - bottom(x)) dx, where a and b are where the curves meet or where you choose to stop. For instance, between y = x and y = x^2 from 0 to 1, the line x sits above the parabola x^2, so the area is integral from 0 to 1 of (x - x^2) dx = 1/2 - 1/3 = 1/6. Because you always subtract bottom from top, the result is a genuine positive area even where both curves dip below the x-axis.

This 'slice and add' move is the gateway to the rest of integration's applications. Spin a region around an axis and the same strips become thin disks or shells, giving you volumes of revolution. Stretch the idea and you get arc length, surface area, the average value of a function, work done by a varying force, and the center of mass — all of them just clever choices of what quantity each thin slice contributes before you integrate.