NAME THE TOOL

A smooth Σ

Coach

Remember Σ? We added a row of strips. Now picture painting a wavy wall with a thin brush, in up-and-down strips. Add up the paint on every strip and you know how much paint the wall took.

Thinner strips fit the wavy edge better. Make them as thin as you like, and the sum becomes smooth. That smooth sum is the integral. For the piggy bank, each strip is one slice of time's coins, and the integral is all the coins added.

Quick check: On a curve, thinner strips make the answer more exact or less exact?

THE SYMBOLS

Where everything goes

b∫a f(x) dx

∫ is a stretched S, for Sum. The stop b sits on top. The start a sits at the bottom. f(x) is the height of each strip (the rate). dx at the end is each strip's thin width.

So the integral adds f(x) · dx over every slice from a to b. It equals the area under f from a to b.

Quick check: In ∫ from 1 to 4 of f(x) dx, which number is the stop?

WORKED EXAMPLE

The bank fills at 4 coins per day from day 0 to day 6. Find the integral.

6∫0 4 dt

Strategy: Read the integral as an area

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

1

Understand

Rate 4 the whole time, from t = 0 to t = 6. We want all the coins added.

Why? The integral is the pile added.

2

Plan

A steady rate makes a rectangle under the graph: height 4, width 6 + (−0).

Why? Area of a rectangle is height × width.

3

Solve

4 × 6 = 24 coins.

Why? Each day's strip holds 4 coins; 6 strips.

4

Look back

As a Σ: 4 + 4 + 4 + 4 + 4 + 4 = 24. ✓

Why? The smooth sum matches the step sum.

Picture way

Symbol way

∫ from 0 to 6 of 4 dt = 4 × 6 = 24

Quick check: True or false: the answer is 4, the rate.

SEE IT

Strips become smooth, and triangles

Thin rectangles under a curve. Make them thinner and their total becomes the area under the curve.

A rate that grows in a straight line from 0 makes a triangle: half of its box. ∫ from 0 to 4 of x dx = half of 4 × 4 = 8.

Trap: Giving the rate, not the pile

∫ from 0 to 5 of 3 dt is 15 coins, not 3.

Trap: Using the stop as the width

∫ from 1 to 4 of 5 dx: the width is 4 + (−1) = 3, so 15, not 20.

Trap: Forgetting the half

∫ from 0 to 4 of x dx is a triangle: 8, not the box 16.

Quick check: ∫ from 2 to 5 of 3 dx: is the width 3 or 5?

WHAT TO LOOK FOR

Clue words and easy mix-ups

∫ … dxAdd rate × dx slices: the area, the pile.
number on topThe stop.
number at the bottomThe start.
f(x), the rate, the heightHow tall each strip is.
the bank started with …The integral leaves the start out. Add it yourself.

Quick check: The bank starts with 4 coins. ∫ from 0 to 3 of 5 dt = 15. How many coins after day 3?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

∫ from 0 to 2 of 4x dx. The height at 2 is 8, the width is 2: 16.

Coach

That is the whole box. The line y = 4x starts at 0, so the area under it is a triangle: half of 16, 8.

Check habit: Sketch the rate first. Flat top: rectangle. Straight line from 0: triangle, half the box.

Quick check: ∫ from 0 to 2 of 4x dx?

TRAP CHECK

Run this before every answer

Stop on top, start at the bottom. Width = stop + (−start).
Sketch the rate: rectangle or triangle?
Answer the area (the pile), not the height (the rate).
If the bank had a start, add it after.