The Trapezoidal Rule

Problem: Find

We put u = x² + 1 and as x = 0 → 1,u = 1 → 2

If u = x² + 1 then du = 2x dx.

But the question does not contain an x dxterm so we cannot solve it using any of the normal integration methods.

We need to use numerical approaches. When software like Mathcad or graphics calculators perform definite integrals, they use numerical methods given earlier.

We can use one of two methods:

• Trapezoidal rule
• Simpson's Rule (in the next section: 6. Simpson's Rule)

The Trapezoidal Rule

We saw the basic idea in our first attempt at solving the area under the arches problem earlier.

Instead of using rectangles as we did in the arches problem, we'll use trapezoids(trapeziums) and we'll find that it gives a better approximation to the area.

Recall that we write "Δx" to mean "a small change in x".

Let's see this in LiveMath. Note that our approximation is much better than using rectangles.

LIVEMath

Now, the area of a trapezoid (trapezium) is given by:

So the approximate area under the curve is found by adding the area of the trapezoids. (Our trapezoids are rotated 90° so that their new base is actually the height. So h = Δx.)

Area ≈

We can simplify this to give us the Trapezoidal Rule, for n trapezoids:

To find Δx for the area from x = a to x = b, we use:

and we also need

y₀ = f(a)

y₁ = f(a + Δx)

y₂ = f(a + 2Δx)
…

y_n = f(b)

---

Note:

• We get a better approximation if we take more trapezoids [up to a limit!].
• The more trapezoids we take, we have: Δx → 0.
• We can write (if the curve is above the x-axis only between x = a and x = b):

---

Exercise: Using n = 5, approximate the integral:

Answer

Here, a = 0 and b = 1.

y₀ = f(a) = f(0) =  = 1
y₁ = f(a + Δx) = f(0.2) = 
y₂ = f(a + 2Δx) = f(0.4) = 
y₃ = f(a + 3Δx) = f(0.6) = 
y₄ = f(a + 4Δx) = f(0.8) = 
y₅ = f(b) = f(1) = 
So the area ≈

So  ≈1.150

Metal Tidbits, Aerospace

matlab task -array and function and matrix and plot

x=[1 2 1/2 -3 -1]

y=[2 0 -3 1/3 2]

x.*y

A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

B=[1 0 -1;-1.5 1.5 -3;1 1 1]

A*B

a=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

d=[0 -3.5 -2; 0 0 1.3;-1.1 -2 0]

c=a+d

d=[3 3;1 2]

dinv=[0.667 -1;-0.333 1]

b=[1;1]

d\b

dinv*b

d/b

d*dinv

M=[1 1 1;1 0 1; -1 0 0]

A=[1 1 1;1 2 1; 1 1 1]

c=M\A

Minv=c/a

A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

B=[1 0 -1;-1.5 1.5 -3;1 1 1]

A|B

c=xor(A, B)

~A

~B

x=[1 -3 3 14 -10 12]

y=[12 6 0 -1 -10 2]

D=x<=y

F=x>=y

K=x==y

G=x~=y

C=[1 2 3 4 10;-22 1 11 -12 4;8 1 6 -11 5;18 1 11 6 4]

compvector=10*ones(4,5)

comp=C==10

D=[7 2 3 10;-2 -3 11 3;8 1 6 5;18 1 11 4]

comp=D==7|D==-3|D==6|D==4

result=D(comp)

t=(0:0.1:10)

in(7*3.142*5*t).*cos(2*3.142*3*t)+exp(-0.1*t)

sin(2*3.142*5.3*t).*sin(2*3.142*5.3*t)

r=[j j+1 j-7 j+1 -3]

r=[j;j+1;j-7;j+1;-3]

k=20*sin(5.3*3.142*2*t)

round(k)

t=(0:0.1:0.5)

k=20*sin(5.3*3.142*2*t)

round(k)

>> x=[1 2 1/2 -3 -1]

x =

  Columns 1 through 3

    1.0000    2.0000    0.5000

  Columns 4 through 5

   -3.0000   -1.0000

>> y=[2 0 -3 1/3 2]

y =

  Columns 1 through 3

    2.0000         0   -3.0000

  Columns 4 through 5

    0.3333    2.0000

> x.*y

ans =

  Columns 1 through 3

    2.0000         0   -1.5000

  Columns 4 through 5

   -1.0000   -2.0000

>> A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

A =

   -1.0000    3.5000    2.0000

         0    1.0000   -1.3000

    1.1000    2.0000    1.9000

>> B=[1 0 -1;-1.5 1.5 -3;1 1 1]

B =

    1.0000         0   -1.0000

   -1.5000    1.5000   -3.0000

    1.0000    1.0000    1.0000

>> A*B

ans =

   -4.2500    7.2500   -7.5000

   -2.8000    0.2000   -4.3000

         0    4.9000   -5.2000

>> a=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

a =

   -1.0000    3.5000    2.0000

         0    1.0000   -1.3000

    1.1000    2.0000    1.9000

>> d=[0 -3.5 -2; 0 0 1.3;-1.1 -2 0]

d =

         0   -3.5000   -2.0000

         0         0    1.3000

   -1.1000   -2.0000         0

>> c=a+d

c =

   -1.0000         0         0

         0    1.0000         0

         0         0    1.9000

>> d=[3 3;1 2]

d =

     3     3

     1     2

>> dinv=[0.667 -1;-0.333 1]

dinv =

    0.6670   -1.0000

   -0.3330    1.0000

>> b=[1;1]

b =

     1

     1

>> d\b

ans =

   -0.3333

    0.6667

>> dinv*b

ans =

   -0.3330

    0.6670

>>d/b

ans=

6

3

>> d*binv

ans =

     6

     3

M=[1 1 1;1 0 1; -1 0 0]

M =

     1     1     1

     1     0     1

    -1     0     0

>> A=[1 1 1;1 2 1; 1 1 1]

A =

     1     1     1

     1     2     1

     1     1     1

>> c=M\A

c =

    -1    -1    -1

     0    -1     0

     2     3     2

>> Minv=c/a

Minv =

    0.1683   -0.0769   -0.7561

   -0.1222   -0.3503   -0.1111

   -0.2144    0.5041    1.6232

>> A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

A =

   -1.0000    3.5000    2.0000

         0    1.0000   -1.3000

    1.1000    2.0000    1.9000

>> B=[1 0 -1;-1.5 1.5 -3;1 1 1]

B =

    1.0000         0   -1.0000

   -1.5000    1.5000   -3.0000

    1.0000    1.0000    1.0000

>> A|B

ans =

     1     1     1

     1     1     1

     1     1     1

> c=xor(A, B)

c =

     0     1     0

     1     0     0

     0     0     0

>> ~A

ans =

     0     0     0

     1     0     0

     0     0     0

> ~B

ans =

     0     1     0

     0     0     0

     0     0     0

>> x=[1 -3 3 14 -10 12]

x =

     1    -3     3    14   -10    12

>> y=[12 6 0 -1 -10 2]

y =

    12     6     0    -1   -10     2

>> D=x<=y

D =

     1     1     0     0     1     0

>> F=x>=y

F =

     0     0     1     1     1     1

>> K=x==y

K =

     0     0     0     0     1     0

>> G=x~=y

G =

     1     1     1     1     0     1

>> C=[1 2 3 4 10;-22 1 11 -12 4;8 1 6 -11 5;18 1 11 6 4]

C =

     1     2     3     4    10

   -22     1    11   -12     4

     8     1     6   -11     5

    18     1    11     6     4

>> compvector=10*ones(4,5)

compvector =

    10    10    10    10    10

    10    10    10    10    10

    10    10    10    10    10

    10    10    10    10    10

>> comp=C==10

comp =

     0     0     0     0     1

     0     0     0     0     0

     0     0     0     0     0

     0     0     0     0     0

result =

    10

>> D=[7 2 3 10;-2 -3 11 3;8 1 6 5;18 1 11 4]

D =

     7     2     3    10

    -2    -3    11     3

     8     1     6     5

    18     1    11     4

> comp=D==7|D==-3|D==6|D==4

comp =

     1     0     0     0

     0     1     0     0

     0     0     1     0

     0     0     0     1

>> result=D(comp)

result =

     7

    -3

     6

     4

>> t=(0:0.1:10)

in(7*3.142*5*t).*cos(2*3.142*3*t)+exp(-0.1*t)

ans =

  Columns 1 through 3

    1.0000    1.2993    0.9825

  Columns 4 through 6

    1.7799    0.9625    1.9512

  Columns 7 through 9

    0.9391    1.7404    0.9139

  Columns 10 through 12

    1.2208    0.8906    0.5843

  Columns 13 through 15

    0.8731    0.0674    0.8633

  Columns 16 through 18

   -0.1391    0.8593    0.0373

  Columns 19 through 21

    0.8561    0.5225    0.8472

  Columns 22 through 24

    1.1243    0.8278    1.6064

  Columns 25 through 27

    0.7970    1.7781    0.7594

  Columns 28 through 30

    1.5679    0.7233    1.0503

  Columns 31 through 33

    0.6981    0.4175    0.6895

  Columns 34 through 36

   -0.0939    0.6972   -0.2940

  Columns 37 through 39

    0.7140   -0.1118    0.7280

  Columns 40 through 42

    0.3776    0.7273    0.9816

  Columns 43 through 45

    0.7051    1.4641    0.6628

  Columns 46 through 48

    1.6355    0.6103    1.4254

  Columns 49 through 51

    0.5630    0.9095    0.5353

  Columns 52 through 54

    0.2805    0.5352   -0.2256

  Columns 55 through 57

    0.5599   -0.4199    0.5969

  Columns 58 through 60

   -0.2326    0.6274    0.2601

  Columns 61 through 63

    0.6342    0.8653    0.6086

  Columns 64 through 66

    1.3473    0.5541    1.5176

  Columns 67 through 69

    0.4864    1.3073    0.4274

  Columns 70 through 72

    0.7931    0.3970    0.1678

  Columns 73 through 75

    0.4049   -0.3330    0.4464

  Columns 76 through 78

   -0.5218    0.5030   -0.3300

  Columns 79 through 81

    0.5494    0.1651    0.5631

  Columns 82 through 84

    0.7705    0.5334    1.2511

  Columns 85 through 87

    0.4663    1.4199    0.3829

  Columns 88 through 90

    1.2092    0.3120    0.6966

  Columns 91 through 93

    0.2786    0.0752    0.2945

  Columns 94 through 96

   -0.4204    0.3523   -0.6038

  Columns 97 through 99

    0.4281   -0.4082    0.4899

  Columns 100 through 101

    0.0885    0.5099

>> sin(2*3.142*5.3*t).*sin(2*3.142*5.3*t)

ans =

  Columns 1 through 3

         0    0.0353    0.1361

  Columns 4 through 6

    0.2883    0.4703    0.6566

  Columns 7 through 9

    0.8207    0.9396    0.9965

  Columns 10 through 12

    0.9833    0.9020    0.7639

  Columns 13 through 15

    0.5886    0.4008    0.2270

  Columns 16 through 18

    0.0917    0.0140    0.0049

  Columns 19 through 21

    0.0656    0.1877    0.3537

  Columns 22 through 24

    0.5404    0.7214    0.8712

  Columns 25 through 27

    0.9686    0.9999    0.9606

  Columns 28 through 30

    0.8564    0.7019    0.5189

  Columns 31 through 33

    0.3332    0.1711    0.0554

  Columns 34 through 36

    0.0024    0.0196    0.1046

  Columns 37 through 39

    0.2453    0.4220    0.6098

  Columns 40 through 42

    0.7820    0.9144    0.9884

  Columns 43 through 45

    0.9935    0.9289    0.8038

  Columns 46 through 48

    0.6359    0.4488    0.2689

  Columns 49 through 51

    0.1216    0.0277    0.0005

  Columns 52 through 54

    0.0437    0.1512    0.3080

  Columns 55 through 57

    0.4919    0.6769    0.8370

  Columns 58 through 60

    0.9495    0.9986    0.9773

  Columns 61 through 63

    0.8887    0.7453    0.5673

  Columns 64 through 66

    0.3797    0.2092    0.0796

  Columns 67 through 69

    0.0094    0.0084    0.0767

  Columns 70 through 72

    0.2048    0.3745    0.5619

  Columns 73 through 75

    0.7406    0.8853    0.9757

  Columns 76 through 78

    0.9990    0.9518    0.8409

  Columns 79 through 81

    0.6820    0.4973    0.3130

  Columns 82 through 84

    0.1551    0.0459    0.0007

  Columns 85 through 87

    0.0260    0.1181    0.2641

  Columns 88 through 90

    0.4434    0.6307    0.7995

  Columns 91 through 93

    0.9261    0.9926    0.989

  Columns 94 through 96

    0.9174    0.7864    0.6150

  Columns 97 through 99

    0.4274    0.2500    0.1079

  Columns 100 through 101

    0.0211    0.0019

>> r=[j j+1 j-7 j+1 -3]

r =

        0 + 1.0000i   1.0000 + 1.0000i  -7.0000 + 1.0000i   1.0000 + 1.0000i  -3.0000         

>> r=[j;j+1;j-7;j+1;-3]

r =

        0 + 1.0000i

   1.0000 + 1.0000i

  -7.0000 + 1.0000i

   1.0000 + 1.0000i

  -3.0000         

>> k=20*sin(5.3*3.142*2*t)

k =

  Columns 1 through 12

         0   -3.7561    7.3785  -10.7384   13.7161  -16.2057   18.1185  -19.3866   19.9648  -19.8324   18.9943  -17.4802

  Columns 13 through 24

   15.3440  -12.6618    9.5290   -6.0570    2.3695    1.4023   -5.1242    8.6638  -11.8950   14.7029  -16.9876   18.6677

  Columns 25 through 36

  -19.6835   19.9988  -19.6024   18.5084  -16.7558   14.4068  -11.5451    8.2726   -4.7057    0.9713    2.7977   -6.4671

  Columns 37 through 48

    9.9063  -12.9930   15.6174  -17.6859   19.1250  -19.8836   19.9345  -19.2760   17.9315  -15.9489   13.3987  -10.3717

  Columns 49 through 60

    6.9755   -3.3312   -0.4318    4.1793   -7.7781   11.1001  -14.0271   16.4549  -18.2971   19.4882  -19.9857   19.7720

  Columns 61 through 72

  -18.8546   17.2663  -15.0635   12.3246   -9.1471    5.6441   -1.9403   -1.8327    5.5404   -9.0509   12.2393  -14.9922

  Columns 73 through 84

   17.2115  -18.8183   19.7554  -19.9895   19.5122  -18.3405   16.5161  -14.1040   11.1899   -7.8776    4.2849   -0.5398

  Columns 85 through 96

   -3.2245    6.8741  -10.2791   13.3182  -15.8834   17.8834  -19.2469   19.9254  -19.8949   19.1564  -17.7361   15.6846

  Columns 97 through 101

  -13.0750   10.0001   -6.5693    2.9047    0.8633

>> round(k)

ans =

  Columns 1 through 20

     0    -4     7   -11    14   -16    18   -19    20   -20    19   -17    15   -13    10    -6     2     1    -5     9

  Columns 21 through 40

   -12    15   -17    19   -20    20   -20    19   -17    14   -12     8    -5     1     3    -6    10   -13    16   -18

  Columns 41 through 60

    19   -20    20   -19    18   -16    13   -10     7    -3     0     4    -8    11   -14    16   -18    19   -20    20

  Columns 61 through 80

   -19    17   -15    12    -9     6    -2    -2     6    -9    12   -15    17   -19    20   -20    20   -18    17   -14

  Columns 81 through 100

    11    -8     4    -1    -3     7   -10    13   -16    18   -19    20   -20    19   -18    16   -13    10    -7     3

  Column 101

     1

>> t=(0:0.1:.5)

t =

         0    0.1000    0.2000    0.3000    0.4000    0.5000

>> k=20*sin(5.3*3.142*2*t)

k =

         0   -3.7561    7.3785  -10.7384   13.7161  -16.2057

>> round(k)

ans =

     0    -4     7   -11    14   -16

x=[1 2 1/2 -3 -1]

y=[2 0 -3 1/3 2]

x.*y

A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

B=[1 0 -1;-1.5 1.5 -3;1 1 1]

A*B

a=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

d=[0 -3.5 -2; 0 0 1.3;-1.1 -2 0]

c=a+d

d=[3 3;1 2]

dinv=[0.667 -1;-0.333 1]

b=[1;1]

d\b

dinv*b

d/b

d*dinv

M=[1 1 1;1 0 1; -1 0 0]

A=[1 1 1;1 2 1; 1 1 1]

c=M\A

Minv=c/a

A=[-1 3.5 2;0 1 -1.3;1.1 2 1.9]

B=[1 0 -1;-1.5 1.5 -3;1 1 1]

A|B

c=xor(A, B)

~A

~B

x=[1 -3 3 14 -10 12]

y=[12 6 0 -1 -10 2]

D=x<=y

F=x>=y

K=x==y

G=x~=y

C=[1 2 3 4 10;-22 1 11 -12 4;8 1 6 -11 5;18 1 11 6 4]

compvector=10*ones(4,5)

comp=C==10

D=[7 2 3 10;-2 -3 11 3;8 1 6 5;18 1 11 4]

comp=D==7|D==-3|D==6|D==4

result=D(comp)

t=(0:0.1:10)

in(7*3.142*5*t).*cos(2*3.142*3*t)+exp(-0.1*t)

sin(2*3.142*5.3*t).*sin(2*3.142*5.3*t)

r=[j j+1 j-7 j+1 -3]

r=[j;j+1;j-7;j+1;-3]

k=20*sin(5.3*3.142*2*t)

round(k)

t=(0:0.1:0.5)

k=20*sin(5.3*3.142*2*t)

round(k)

plot with matlab

x = 0:1:15;

y1=(14.*(x.^2))+(3.*(x)-(25));

y2=(-8.*(x.^2))+(7.*(x)+(30));

[AX,H1,H2] = plotyy(x,y1,x,y2,'plot');

>> x=[1:1:15];

y1=(14.*(x.^2))+(3.*(x)-(25));

(y1(x))

ans =

Columns 1 through 7

-8          37         110         211         340         497         682

Columns 8 through 14

895        1136        1405        1702        2027        2380        2761

Column 15

3170

>> x=[1:1:15];

y2=(-8.*(x.^2))+(7.*(x)+(30));

(y2(x))

ans =

Columns 1 through 7

29          12         -21         -70        -135        -216        -313

Columns 8 through 14

-426        -555        -700        -861       -1038       -1231       -1440

Column 15

-1665

x=-pi:0.1:pi; y1=4*sin(x); x1=4*cos(x); plot(x1,y1); hold on x=-pi:0.1:pi; y2=2*sin(x); x2=2*cos(x); plot(x2,y2); hold on z=-2.8:0.1:2.8; q=-2.8:0.1:2.8; y3=2.8; x3=-2.8; plot(y3,q,'r'); hold on plot(x3,q); hold on plot(z,y3); hold on plot(z,x3)

 

n = 23;

t = -pi:0.5:pi;

x = (n.*t)-(sin(n.*t));

plot(t,x)

x =

Columns 1 through 8

-72.2566  -61.6321  -50.1029  -37.6991  -25.3548  -13.9424   -3.3714    7.3182

Columns 9 through 13

18.9639   31.4151   43.6888   54.9855   65.5153

n = 23; t = -pi:0.5:pi; y = (n.*sin(t))./(cos(n.*t)); stem(t,y) box y =   Columns 1 through 8     0.0000   22.8154  -36.3225  -22.9804  -48.3917   23.7080    3.2674  -21.2528   Columns 9 through 13

 

 

   27.7861   22.8221   67.6937  -24.2122   -6.6005