Cambridge Additional Mathematics

412 Integration (Chapter 15)

n AL AU Average

4 0 :218 75 0 :468 75 0 :343 75

10 0 :285 00 0 :385 00 0 :335 00

25 0 :313 60 0 :353 60 0 :333 60

50 0 :323 40 0 :343 40 0 :333 40

The table alongside summarises the results you should obtain

for n=4, 10 , 25 , and 50.

The exact value ofAis in fact^13 , as we will find later in the

chapter. Notice how bothALandAU are converging to this

value asnincreases.

EXERCISE 15A.1

1 Consider the area between y=x and thex-axis from x=0to x=1.

a Divide the interval into 5 strips of equal width, then estimate the area using:

i upper rectangles ii lower rectangles.

b Calculate the actual area and compare it with your answers ina.

2 Consider the area between y=

1

x

and thex-axis from x=2to x=4. Divide the interval into

6 strips of equal width, then estimate the area using:

a upper rectangles b lower rectangles.

3 Use rectangles to find lower and upper sums for the area between the graph of y=x^2

and thex-axis for 16 x 62. Use n=10, 25 , 50 , 100 , and 500. Give your answers

to 4 decimal places.

Asngets larger, bothALandAU converge to the same number which is a simple

fraction. What is it?

4aUse lower and upper sums to estimate the area between each of the following functions and the

x-axis for 06 x 61. Use values of n=5, 10 , 50 , 100 , 500 , 1000 , and10 000. Give your

answer to 5 decimal places in each case.

i y=x^3 ii y=x iii y=x

1

(^2) iv y=x
1
3
b For each case ina,ALandAU converge to the same number which is a simple fraction. What
fractions are they?
c Using your answer tob, predict the area between the graph of y=xa and thex-axis for
06 x 61 and any number a> 0.
5 Consider the quarter circle of centre (0,0) and
radius 2 units illustrated.
Its area is^14 (full circle of radius2)
=^14 £¼£ 22
=¼units^2
a Estimate the area using lower and upper rectangles for n=10, 50 , 100 , 200 , 1000 , and10 000.
Hence, find rational bounds for¼.
b Archimedes found the famous approximation 31071 <¼< 317.
For what value ofnis your estimate for¼better than that of Archimedes?
AREA
FINDER
2
2 x
y
y = ~#4 #- #x 2
O
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(^05255075950525507595)
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\412CamAdd_15.cdr Monday, 7 April 2014 3:57:31 PM BRIAN