The Chemistry Maths Book, Second Edition

5 Integration

5.1 Concepts

Consider a body moving along a curve from point Aat timet 1 = 1 t

A

to point Bat time

t 1 = 1 t

B

. Let the distance from A along the curve at some intermediate time tbes(t), as

illustrated in Figure 5.1.

If v(t)is the velocity along the curve at time tthen, by the discussion of Section 4.11,

v(t) 1 = 1 ds 2 dtis the gradient of the graph of the function s(t). Therefore, if s(t)is a

known function, v(t)is obtained by differentiation. Conversely, if v(t)is a known

function thens(t)is that function whose derivative is v(t); that is, to finds(t)we

need to reversethe differentiationds 2 dt 1 = 1 v(t).

The distance travelled between two points is equal to the average velocity multiplied

by the time taken. For the motion shown in Figure 5.1, the distance ABalong the curve

is therefore

(5.1)

where is the average (or mean) value of v(t)between Aand B. For example, if

the body undergoes constant acceleration from v 1 = 1 v

A

at Ato v 1 = 1 v

B

at B, as

illustrated in Figure 5.2,

1

the average velocity is and the total

distance travelled is

dt=+×−t

1

2

()()vv

AB BA

vvv=+()

AB

2

v

dtt=× −v ()

BA

1

This graph of velocity as a function of time for a body moving with uniform acceleration appeared in

Quaestiones super geometriam Euclidesby Nicole Oresme (c.1323–1382), Dean of Rouen Cathedral and Bishop of

Lisieux. This was possibly the first graph of a variable physical quantity. Oresme also considered the extension of

his ‘latitude of forms’ to the representation of the ‘quality’ of a surface by a body in three dimensions and ‘the quality

of a body will no doubt be represented by something having four dimensions in a different kind of quantity’.

A

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Figure 5.1