Water flows through a valve at the rate gallons per minute — positive means filling, negative means draining. The tank starts with 20 gallons. Drag time forward and watch the shaded area turn into gallons: blue area above the axis adds water, red area below it takes water away.
A(t) = ∫₀ᵗ r — the running net change, traced as a curve of its own. It climbs while r > 0, crests exactly where r = 0, and falls while r < 0. Look at the little tangent tick riding its tip: its slope at every moment is r(t). Hold that thought — it's the whole next lesson.
The tank's amount is the starting 20 gal plus the signed area so far.
Before the shortcuts, the recipe. Every Riemann approximation is the same four moves: slice the interval, pick a point in each slice, evaluate f there for the height, and add up height × width. Here it is for on [0, 4] with n = 4 — click step to lay each rectangle yourself and watch the sum grow a term at a time.
Now that you can build one by hand, scale it. Pick a function and a method, then drag n — the number of rectangles — and watch the estimate close in on the true area. The pattern of over and under isn't random: it's decided by whether the curve is rising, falling, or bending.
On the AP exam you'll rarely be handed a formula — you'll be handed a table, with unequal spacing, and asked to estimate the integral. The widths come from the t-row, the heights from the r-row. Pick a method and watch the sum assemble itself, term by term.
| t (min) | 0 | 2 | 5 | 7 | 10 |
|---|---|---|---|---|---|
| R(t) (gal/min) | 4.2 | 6.0 | 7.4 | 5.6 | 3.0 |
A Riemann sum and a definite integral are the same idea at two zoom levels. The integral sign is a stretched S — a sum — and every piece of the sum survives in the notation.
Integrating a rate gives the net change of the quantity. Add the starting amount to get the new amount.
Rising or falling decides Left vs Right; bending decides Midpoint vs Trapezoid — and those two always land on opposite sides.
| f increasing | f decreasing | |
|---|---|---|
| Left | under | over |
| Right | over | under |
| concave up | concave down | |
| Midpoint | under | over |
| Trapezoid | over | under |
Every Riemann sum hides the same three ingredients — find them and you can convert in either direction.
A rectangle's area is height × width, so the integral's units are (units of f) × (units of x) — gallons/min × min = gallons.
Five checks, straight from the exam's playbook. Try each before you reveal the worked answer.
Eight fast multiple-choice checks across the whole lesson. Pick an answer to see whether it's right and why.
Tables are usually spaced unevenly on purpose. Every width is the difference of consecutive t-values — compute each one; never divide the interval length by the number of columns.
A left sum uses every value except the last; a right sum uses every value except the first. Using all five values of a five-column table means you've double-dipped.
Don't memorize blindly — picture it. Left rectangles tuck under a rising curve because each one's height is set at its lowest point. Re-derive the rule from the picture every time.
∫r dt is net change — negative rate subtracts. If a question asks for the total amount pumped in and out, that's ∫|r| dt, a different number.
The integral gives the change. “How much is in the tank at t = 10?” needs initial amount + net change — leaving off the +20 is a classic dropped point.
On an FRQ, an estimate of “56.2” loses a point that “56.2 gallons” earns. The units are part of the answer, not decoration.