Finding the area of curved

irregular shapes

LEAVING CERTIFCATE ORDINARY & HIGHER LEVELS

Find the area of your shape

1A

Work as accurately as

you can in centimetres!

You found the area of an A4 page!

1A

1B

Trapezoidal Rule

2 m

2.4 m

(a) Find the area of the stained glass window.(b) Draw a scaled diagram of the window using 1 m = 5 cm on square paper.(c) Divide the shape up into:

(i) 4 trapezium's (ii) 5 trapezium’s(ii) 10 trapeziumsand calculate the area of the window

each time, (using the trapezium’s).(d) Which part produced the smallest error and why?

Formula and Tables Page 12

What do all the terms in the formula mean?

𝐴=𝑇1+𝑇 2+𝑇3+𝑇 4+⋯+𝑇 𝑛

Derivation of the formula

𝒂 𝒃𝒉 𝐴=12

(𝑎+𝑏 )h

Trapezium Rule

Why Trapezoidal Rule rather than Simpson’s Rule?

1. Easier to derive and understand what is going on.

2. Area of a trapezium is on Strand 3 for Ordinary Level students from 2015.(N.B. It is on for Foundation Level in 2014)

3. For many practical cases it (the Trapezoidal rule) is accurateenough.

2002

2013 HL

Using 5 intervals approximate the following integral .

1 6 2 3 4 5

21

22

23

24

21 5

26

1 2

0

1 2

0

2( )2

1 (0) 1

1 (0.2) 0.9797

1 (0.4) 0.9165

1 (0.6) 0.8

1 (0.8) 0.6

1 (1) 0

0.21 1 0 2 0.9797 0.9165 0.8 0.6

2

1 0.75924

n

n

hA y y y y y y

y

y

y

y

y y

y y

x dx

x dx

𝐴≈h2 [𝑦 1+𝑦𝑛+2 ( 𝑦2+𝑦3+𝑦 4+⋯+𝑦𝑛− 1 ) ]

Other Methods

0

1

√1−𝑥2𝑑𝑥