NAME THE TOOL

Pile up twice

Coach

Remember the integral? It piled up along one line: slice, slice, slice. That is like cutting a chocolate bar one way.

Now pour sand over a whole floor. First pile up the sand along one row. Then pile up the rows. Two pile-ups: a double integral. A flat sheet of chocolate is cut across and down: two directions.

Quick check: A block of cheese cut into tiny cubes needs cuts in 2 directions or 3 directions?

THE SYMBOLS

Inside first, then outside

3∫0 4∫0 2 dy dx

Read it from the inside out. The inside ∫ from 0 to 4 goes with dy: it piles up along a row of length 4. The outside ∫ from 0 to 3 goes with dx: it piles up the 3 rows.

∬ is short for this double pile-up. ∭ (triple) piles up in three directions. This card is the idea and the notation only.

Quick check: In that double integral, which bounds go with dy?

WORKED EXAMPLE

Sand 3 deep over a 2 by 5 floor

2∫0 5∫0 3 dy dx

Strategy: Inside pile-up, then outside pile-up

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

1

Understand

Depth 3 everywhere. Rows of length 5. 2 rows.

Why? Steady depth, rectangle floor.

2

Plan

Inside: one row holds 3 × 5. Outside: add 2 rows.

Why? Accumulate twice.

3

Solve

One row: 15. Two rows: 30.

Why? Row, then rows.

4

Look back

Floor area 2 × 5 = 10, times depth 3 = 30. ✓

Why? Same answer the other way round.

Row way

15 per row, 2 rows: 30.

Floor way

3 × (2 × 5) = 30

Quick check: True or false: the answer is 10, the floor area.

SEE IT

Row by row

Trap: Accumulating only once

The floor area is 12. The sand is 2 deep, so 24.

Trap: Stopping after one row

One row is 8. Three rows are 24.

Trap: Adding the sizes

2 + 3 + 4 = 9 is not a pile of sand.

Quick check: Sand 2 deep, 3 rows of length 4. Total?

WHAT TO LOOK FOR

Clue words and easy mix-ups

∬ or ∫∫Pile up twice.
∭Pile up three times: across, down and up.
dy dxInside ∫ with dy, outside ∫ with dx.
a hole, digging outNegative depth: the total is negative.

Quick check: A hole 3 deep over a 1 by 5 floor. Sand total?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

Sand 4 deep over a 2 by 3 floor. Each row of 3 holds 4 × 3 = 12. Done: 12.

Coach

That was one row. There are 2 rows: 24.

Check habit: Count the rows before you stop.

Quick check: Total?

TRAP CHECK

Run this before every answer

Pair each ∫ with its d: inside with inside.
Inside pile-up: one row.
Outside pile-up: all the rows.
Holes are negative.