5.5 Numerical Integration

Mt. Shasta, CaliforniaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998

Using integrals to find area works extremely well as long as we can find the antiderivative of the function.

Sometimes, the function is too complicated to find the antiderivative.

At other times, we don’t even have a function, but only measurements taken from real life.

What we need is an efficient method to estimate area when we can not find the antiderivative.

211

8y x

43

0

1

24A x x

4 2

0

11

8A x dx

0 4x

Actual area under curve:

20

3A 6.6

211

8y x 0 4x

Left-hand rectangular approximation:

Approximate area: 1 1 1 31 1 1 2 5 5.75

8 2 8 4

(too low)

Approximate area: 1 1 1 31 1 2 3 7 7.75

8 2 8 4

211

8y x 0 4x

Right-hand rectangular approximation:

(too high)

Averaging the two:

7.75 5.756.75

2

1.25% error (too high)

Averaging right and left rectangles gives us trapezoids:

1 9 1 9 3 1 3 17 1 171 3

2 8 2 8 2 2 2 8 2 8T

1 9 1 9 3 1 3 17 1 171 3

2 8 2 8 2 2 2 8 2 8T

1 9 9 3 3 17 171 3

2 8 8 2 2 8 8T

1 27

2 2T

27

4 6.75 (still too high)

Trapezoidal Rule:

0 1 2 12 2 ... 22 n n

hT y y y y y

( h = width of subinterval )

This gives us a better approximation than either left or right rectangles.

1.031251.28125

1.781252.53125

211

8y x 0 4x

Compare this with the Midpoint Rule:

Approximate area: 6.625 (too low)0.625% error

The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

Midpoint Rule: 6.625 (too low)0.625% error

Trapezoidal Rule: 6.750 1.25% error (too high)

Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction.

If we use a weighted average:

2 6.625 6.7506.6

3

This is the exact answer!

Oooh!

Ahhh!

Wow!

1x 2x 3x 4xh h h h

This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules.

Midpoint:

1 3 1 32 2 2M h y h y h y y

Trapezoidal:

0 2 2 4

1 12 2

2 2T y y h y y h

0 2 2 4T h y y h y y

0 2 42T h y y y

1 3 0 2 4

14 2

3h y y h y y y

twice midpoint trapezoidal

1 3 0 2 44 4 23

hy y y y y

0 1 2 3 44 2 43

hy y y y y

2

3

M T

Simpson’s Rule:

0 1 2 3 2 14 2 4 ... 2 43 n n n

hS y y y y y y y

( h = width of subinterval, n must be even )

Example: 211

8y x 1 9 3 17

1 4 2 4 33 8 2 8

S

1 9 171 3 3

3 2 2

120

3 6.6

Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly.

Simpson’s rule will usually give a very good approximation with relatively few subintervals.

It is especially useful when we have no equation and the data points are determined experimentally.