Speak Maths · Last school years

Adding up slices: integrals

An integral adds up a great many tiny pieces. It finds areas, and totals that build up as things change.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$\int f (x) d x$ Say: the integral of f of x, d xMeans: a function whose derivative is f: undo differentiating
$\int_{a}^{b} f (x) d x$ Say: the integral from a to b of f of x, d xMeans: everything f adds up between a and b: the area under its graph
$F (b) - F (a)$ Say: F of b minus F of aMeans: how much F has gained from a to b, where F is the undo of f
$[F (x)]_{a}^{b}$ Say: F of x, evaluated from a to bMeans: a short way to write F of b minus F of a
$f (x) d x$ Say: f of x, d xMeans: a thin slice: height f of x, width d x, as thin as you like
$+ C$ Say: plus CMeans: an unknown constant: many functions share one derivative

The story: Filling the bath

Subtitles:

Tap a line to see its words.

$r (t) = 2 t$ A tap is opened slowly. At minute t it pours r of t litres per minute, equal to 2 t.
$r (t) d t = 2 t d t$ In a thin slice of time d t, the tap pours r of t times d t litres.
$V = \int_{0}^{5} r (t) d t$ Add all the slices from minute 0 to minute 5. That total, V, is an integral.
$\int 2 t d t = t^{2} + C$ Undo the differentiating: the integral of 2 t, d t, is t squared plus C.
$F (t) = t^{2} V = [F (t)]_{0}^{5} = F (5) - F (0) = 25$ Let F of t be t squared. Then V is F of 5 minus F of 0, which is 25 litres.
$\frac{1}{2} \cdot 5 \cdot 10 = 25$ Check with the graph: a triangle 5 minutes wide and 10 tall. Half of 5 times 10 is 25.
$\int_{0}^{3} 2 t d t = [t^{2}]_{0}^{3} = 9$ After 3 minutes, the integral from 0 to 3 of 2 t, d t, gives 9 litres.
$\int_{0}^{6} 2 t d t = [t^{2}]_{0}^{6} = 36$ The bath holds 36 litres. The integral from 0 to 6 gives 36, so it is full after 6 minutes.
$\frac{9}{36} = \frac{1}{4}$ At half the time, 3 of 6 minutes, the bath has only a quarter of the water.

In words this story is 704 letters long; in symbols, 122.

Read it

What does it say?

$\int_{0}^{4} f (x) d x$

$[F (x)]_{1}^{3}$

$\int 3 x^{2} d x$

$f (x) d x$

$\int_{2}^{6} v (t) d t$

$\int x^{2} d x = \frac{x ^{3}}{3} + C$

$[sin x]_{0}^{π}$

$\int_{5}^{1} f (x) d x = - \int_{1}^{5} f (x) d x$

$A = \int_{0}^{3} (x^{2} + 1) d x$

$3 d t$

$F (10) - F (0) = 50$

$\int e^{x} d x = e^{x} + C$

Write it

Which notation says this?

The area under f from 1 to 5.

How much F gains from a to b.

The integral of 2 x is x squared plus a constant.

All the rain that fell between hour 2 and hour 8, when it fell at r of t millimetres per hour.

A thin slice of area under the graph of g: height g of x, width d x.

F of x, evaluated from 0 to 4.

All the functions whose derivative is 4 x cubed.

The area under f from a to b is F of b minus F of a, where F is an antiderivative of f.

The integral from 0 to 1 of 2 x, d x, is 1.

Square x, put in 3, then take away what you get putting in 1.

The integral of 1 over x is the natural logarithm of the absolute value of x, plus a constant.

A tap runs at a steady 3 litres per minute. The water poured from minute 0 to minute 5 is 15 litres.

Build it

Tap the tiles in order. One tile does not belong.

The integral from 0 to 2 of 3 x squared, d x, equals 8.

F of 4 minus F of 1 equals the integral from 1 to 4 of f of x, d x.

The integral of x squared, d x, is x cubed over 3 plus C.

The integral from 1 to 4 of 2 x, d x, equals 15.

F of t, evaluated from 0 to 6, equals F of 6 minus F of 0.

The volume V is the integral from 0 to 10 of r of t, d t.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Which one is the antiderivative?

Here f(x) = 2x, g(x) = 3x², h(x) = 1, and F₁ = x² + 1, F₂ = x³ − 4, F₃ = x² + x, F₄ = x + 9. Judge each sentence: true or false?

	$F_{1}$	$F_{2}$	$F_{4}$
$f$	✓
$g$		✓
$h$			✓

A tick means the column is an antiderivative of the row: its slope is the row’s function.

$\int f d x = F_{1} + C$
$\int g d x = F_{1} + C$
$\int_{0}^{2} g d x = F_{2} (2) - F_{2} (0)$
$[F_{1} (x)]_{0}^{3} = \int_{0}^{3} f d x$
$\int_{0}^{1} h d x = F_{3} (1) - F_{3} (0)$
$\int h d x = F_{4} + C$

2 Five ways to write a tap

A tap pours 2t litres per minute. Pair each line with what it says.

Tap a line, then its meaning.

3 The bracket trick

The line should show how to compute the integral of 2x from 0 to 3. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 The missing constant

Which answer is the integral without limits?

$\int 5 d x$

5 Top minus bottom

Evaluate the bracket.

$[x^{3}]_{1}^{2}$

6 Fill the bath

The tap pours 2t litres per minute at minute t. Write in symbols that the water collected in the first 6 minutes is 36 litres. Then compare.

Your version

7 All the answers at once

Every function whose slope is cos x is sin x plus some constant. Write that in symbols. Then compare.

Your version

8 Four hours of rain

Rain fell at a steady rate r during each hour, as the grid shows (millimetres per hour). Judge each sentence: true or false?

	$r = 0$	$r = 2$	$r = 5$
$0 \leq t < 1$		✓
$1 \leq t < 2$			✓
$2 \leq t < 3$			✓
$3 \leq t < 4$	✓

A tick means that during the row’s hour the rain fell at the column’s rate r.

$\int_{0}^{4} r (t) d t = 12$
$\int_{1}^{3} r (t) d t = 10$
$\int_{0}^{1} r (t) d t = \int_{3}^{4} r (t) d t$
$\int_{3}^{4} r (t) d t = 0$
$\int_{0}^{2} r (t) d t = 10$
$\int_{4}^{0} r (t) d t = - 12$

9 Five integrals that look alike

Same sign, small changes. Pair each with what it is.

Tap a line, then its meaning.

10 The constant that got lost

This should give every function whose derivative is 3x². One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

11 Top and bottom

The line should compute the integral of 2x from 1 to 3. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

12 Reading the odometer

A cart’s speed is v = 3t². How far does it go in the first 2 seconds?

$v (t) = 3 t^{2}, \int_{0}^{2} v (t) d t = ?$

13 A bracket with cosines

Evaluate the bracket.

$[cos x]_{0}^{π} = ?$

14 Pin down the constant

F is an antiderivative of 2x, and F(1) = 5. Which function is F?

$\int 2 x d x = F (x), F (1) = 5, F (x) = ?$

15 Distance from speed

Write in symbols: the distance travelled from time a to time b is the integral of the speed, and equals the change in position. Then compare.

Your version

16 Two rules of the bounds

Write in symbols: an integral over a stretch of no width is zero, and swapping the bounds flips the sign. Then compare.

Your version

Tell your own

Tell a three-line story about a total that builds up while a rate changes, using an integral and F of b minus F of a. Then read it aloud.

Kept on this device only.

Adding up slices: integrals

The alphabet

The story: Filling the bath

Read it

Write it

Build it

Challenges

1 Which one is the antiderivative?

2 Five ways to write a tap

3 The bracket trick

4 The missing constant

5 Top minus bottom

6 Fill the bath

7 All the answers at once

8 Four hours of rain

9 Five integrals that look alike

10 The constant that got lost

11 Top and bottom

12 Reading the odometer

13 A bracket with cosines

14 Pin down the constant

15 Distance from speed

16 Two rules of the bounds

Tell your own

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