NAME THE TOOL

A recipe or a cake?

Coach

Remember the integral? With a start and a stop, it gave a number: the coins added from day 0 to day 5.

Now leave off the start and the stop. The question changes to: which pile grows at this rate? You get a formula, like a recipe. But the recipe is missing one amount: how many coins were in the bank at the very start. We write that unknown start as + C.

Quick check: An integral has no start and no stop. Do you get a number or a formula?

THE SYMBOLS

Two kinds of ∫

Indefinite (no bounds): ∫ 2x dx = x² + C. Here x² means x · x. The pile x² grows at rate 2x (Card 0.7: the rate of t · t is 2t). C is the start.

Definite (bounds): a number. It is the area, or the pile at the stop + (−pile at the start). The C cancels, because both piles have the same start.

rate 3 (any steady number)pile 3x + C
rate 2xpile x² + C
rate 0pile C: nothing changes

Quick check: ∫ 3 dx = ?

WORKED EXAMPLE

Find ∫ 2x dx and ∫ from 0 to 4 of 2x dx

Strategy: Bounds or no bounds? Then recipe or cake

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

The first has no bounds: a formula. The second has bounds: a number.

Why? Look for numbers on the ∫.

2

Plan

Which pile grows at rate 2x? x². Then use it, or the area, for the number.

Why? The rate tells us the pile.

3

Solve

∫ 2x dx = x² + C. With bounds: 16 + (−0) = 16.

Why? Pile at 4 + (−pile at 0).

4

Look back

Area way: a triangle under 2x, half of 4 × 8 = 16. ✓

Why? Two ways give 16.

Area way

Triangle under 2x from 0 to 4: half of 4 × 8 = 16.

Formula way

x² at 4 + (−x² at 0) = 16 + (−0) = 16

Quick check: True or false: ∫ 2x dx (no bounds) equals 16.

SEE IT

Why C cancels

Bank A starts with 0 coins, bank B with 5. Both grow at rate 2x. Piles: x² and x² + 5. From x = 4 to x = 6, A gains 36 + (−16) = 20 and B gains 41 + (−21) = 20. Same gain, whatever the start.

Trap: Forgetting + C

∫ 2x dx is x² + C, not just x².

Trap: Using only the top bound

∫ from 4 to 6 of 2x dx is 36 + (−16) = 20, not 36.

Trap: Giving the rate

∫ 2x dx is not 2x + C. 2x is the rate; the pile is x² + C.

Quick check: ∫ from 4 to 6 of 2x dx: 20 or 36?

WHAT TO LOOK FOR

Clue words and easy mix-ups

∫ with no numbersIndefinite: formula + C.
∫ from a to bDefinite: a number.
the bank started with …That is the C.
which pile grows at rate …Indefinite.

Quick check: ∫ from 1 to 4 of 6 dx. Number or formula?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

∫ 4 dx. The pile that grows at 4 a day is 4x. Done: 4x.

Coach

Which 4x? A bank could start with 0 coins, or 7, or 20 and still grow at 4 a day. I need + C: 4x + C.

Check habit: No bounds? Write + C before you move on.

Quick check: ∫ 4 dx = ?

TRAP CHECK

Run this before every answer

Bounds or no bounds?
No bounds: the pile formula, + C.
Bounds: a number. Area, or pile at stop + (−pile at start).
Rate in, pile out: do not hand back the rate.