19/06/2007 VER2-0

(a) Find the Area bounded by the curve y=20sin200t and the t axis. Between t=0 and t=30ms

EXERCISE SET 6 – Area under a Curve & Mean

005.0

0

005.0

0 200

200cos120200sin206Area

t

dtt

We can see our curve crosses thex axis at: 0, 5, 10, 15, 20, 25 & 30ms

-25

-20

-15

-10

-5

0

5

10

15

20

25

10 20 30 40 50 60

Time t ms

Am

pli

tud

e

We could break this problem into 6 definite integrals.

However we can use symmetry to help solve the problem.

Just solve one definite integral & multiply by 6.

005.0

0200cos

5

3Area t

]0[cos]005.0200[cos

5

3

1cos5

3Area

11

5

3

units382.0

5

6

19/06/2007 VER2-0

(b) Find the area bounded by the curves y=x3 and y=x.

EXERCISE SET 6 – Area under a Curve & Mean

0

1

420

1

31 )

42(

xx

dxxxA

Find where the curves intersect.ie solve the equation. x3 -x=0

x(x2-1)=x(x-1)(x+1)=0 ie x = -1, 0, +1

Note we could have used symmetry to help solve this problem.

4

1

4

1

4

1

4

1Area 21 AA

-2

-1

0

1

2

3

4

5

-1 0 1 2 3

y=x3

y=x

4

1]

4

)1(

2

)1([]00[

42

1

A

1

0

421

0

32 )

42(

xxdxxxA 4

1]00[]

4

1

2

1[

42

2unit 2

1

19/06/2007 VER2-0

(c) Find the mean of y=x2+2x+1 between x=0 & x=4

EXERCISE SET 6 – Area under a Curve & Mean

ab

dxxfb

a

)(

Mean 04

12

24

0

dxxx

4

0

23

]3

[4

1 xx

x

]443

4[

4

1 2

3

53

16]14

3

4[

2

333.103

110

)]003

0()44

3

4[(

4

1 2

32

3