NAME THE TOOL

Coins in, coins out

Coach

Remember the integral as an area? So far every rate was positive: coins going in.

Some days the panda takes coins out. Then the rate is negative, and its bar hangs below the axis. That area counts as negative, because it shrinks the pile.

Quick check: The panda takes out 2 coins a day for 3 days. Does the pile go up or down?

THE SYMBOLS

∫ keeps the sign

∫ = (area above) + (−area below)

The integral adds rate × dx slices. When the rate is negative, each slice adds a negative. So the area below the axis comes in with a minus sign.

A rate of −2 for 3 days: ∫ from 0 to 3 of (−2) dt = −6.

Quick check: ∫ from 0 to 3 of (−2) dt: 6 or −6?

WORKED EXAMPLE

+3 a day for 4 days, then −2 a day for 3 days. Total change?

Strategy: Signed pieces, then add

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

1

Understand

Two pieces: one above the axis, one below.

Why? In and out, so watch the signs.

2

Plan

Each piece: rate × time, with its sign. Then add.

Why? The integral adds signed slices.

3

Solve

3 × 4 = 12. (−2) × 3 = −6. 12 + (−6) = 6.

Why? The withdrawals count as negative.

4

Look back

Pile from 0: 3, 6, 9, 12, then 10, 8, 6. ✓

Why? Counting day by day agrees.

Area way

Symbol way

3 × 4 + (−2) × 3 = 12 + (−6) = 6

Quick check: True or false: the total is 18, because 12 + 6 = 18.

SEE IT

Balanced pieces

∫ from −2 to 2 of x dx: a triangle of area 2 below and a triangle of area 2 above. Signed total: −2 + 2 = 0.

Trap: Dropping the sign

A rate of −2 for 3 days gives −6, not 6.

Trap: Counting area below as positive

+4 for 2 days then −1 for 3 days: 8 + (−3) = 5, not 11.

Quick check: ∫ from −2 to 2 of x dx: 0 or 4?

WHAT TO LOOK FOR

Clue words and easy mix-ups

takes out, spends, loses, below the axisNegative rate, negative area.
puts in, gains, above the axisPositive.
total change, ∫Add the pieces with their signs.
how much area is shadedThat would count every piece as positive. The integral does not.

Quick check: A rate of −4 for 2 days. Signed area?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

+7 a day for 2 days, then −3 a day for 3 days. Areas 14 and 9. Total 23.

Coach

The second piece took coins out. 14 + (−9) = 5.

Check habit: Shade the pieces below the axis in a different color, and give each a minus sign.

Quick check: Total?

TRAP CHECK

Run this before every answer

Which pieces are above the axis, and which are below?
Each piece: rate × time, keeping its sign.
Add them.
If the bank had a start, add it last.