AP Calculus
Integrals in Desmos
Definite integrals are signed area. Desmos can shade the region, compute the value, and help you see when the net area is smaller than the total area.
Shade a definite integral
Graph f(x), then compute ∫(f(x), a, b). To shade, use an inequality such as 0 ≤ y ≤ f(x){a ≤ x ≤ b} when f is above the axis. For a function that dips below, shade in two pieces so you can see cancellation.
Net area vs. total area
Net area is ∫ f. Total area is ∫ |f|. If a FRQ asks for area of a region, it wants the geometric area, which is always positive. Split at zeros of f and add the absolute values of each piece.
Riemann sums
Left, right, and midpoint sums are easy to sketch with rectangles of width Δx = (b − a)/n. Increase n with a slider to watch the sum approach the integral. This is the definition, not a trick.