Welcome to Numerical Integration!

Ever tried to solve a math problem and realized there isn't a neat, perfect answer? In your standard A Level Maths, most integration problems are designed to be "nice." But in the real world, functions can be messy, and sometimes an exact integral is impossible to find using standard rules.
That’s where Numerical Integration comes in! We use clever rules to estimate the area under a curve. Think of it like trying to measure the area of a puddle—you can't use a simple ruler, but you can break it down into shapes you do know. In this chapter, we’ll look at the Midpoint Rule, the Trapezium Rule, and the powerhouse of the group: Simpson’s Rule.

1. The Basics: What are we doing?

The goal is to estimate the definite integral \( \int_{a}^{b} f(x) dx \). We divide the interval from \( a \) to \( b \) into \( n \) equal strips, each with a width of \( h \).

The Formula for Strip Width:
\( h = \frac{b - a}{n} \)

Analogy: Imagine you are tiling a floor. \( a \) and \( b \) are the walls, and \( h \) is the width of each tile. If you want more tiles (\( n \)), each tile must be thinner (\( h \)).

Quick Review: Prerequisite Concepts

  • Ordinate (\( y \)): The value of the function at a specific \( x \) point.
  • Interval: The space between our start point \( a \) and end point \( b \).
  • Radians: Always make sure your calculator is in Radians mode for calculus!

Key Takeaway: Numerical integration is about approximating area by breaking a curve into smaller, manageable strips.

2. The Midpoint Rule (\( M_n \))

The Midpoint Rule estimates the area of each strip by using a rectangle. But instead of using the height at the start or end of the strip, we use the height at the middle.

The Formula:
\( M_n = h(y_{1/2} + y_{3/2} + \dots + y_{n-1/2}) \)

Here, \( y_{1/2} \) simply means the value of the function at the midpoint of the first strip.

Step-by-Step:
1. Calculate the width \( h \).
2. Find the midpoints of each strip.
3. Plug those midpoints into your function \( f(x) \) to get the heights (\( y \) values).
4. Add the heights together and multiply the total by \( h \).

Common Mistake: Don't confuse the number of strips (\( n \)) with the number of midpoints. If you have 4 strips, you will have 4 midpoints!

3. The Trapezium Rule (\( T_n \))

You might remember this from A Level Maths! Instead of flat-top rectangles, we use trapeziums to "follow" the slope of the curve.

The Formula:
\( T_n = \frac{1}{2}h \{ (y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1}) \} \)

Memory Aid: "Ends stay once, middles twice." You use the very first and very last heights once, but every height in between gets counted twice because it's shared by two trapeziums.

Key Takeaway: \( T_n \) is usually better than simple rectangles, but it can still struggle with very curvy lines.

4. The Powerhouse: Simpson’s Rule (\( S_{2n} \))

If the Trapezium rule is a "connect-the-dots" with straight lines, Simpson’s Rule is like connect-the-dots with parabolas. It is much more accurate!

The Formula:
\( S_{2n} = \frac{1}{3}h \{ (y_0 + y_{2n}) + 4(y_1 + y_3 + \dots + y_{2n-1}) + 2(y_2 + y_4 + \dots + y_{2n-2}) \} \)

The Weighting Pattern: 1, 4, 2, 4, 2... 4, 1.
The "ends" are multiplied by 1. The "odd-numbered" ordinates are multiplied by 4, and the "even-numbered" ordinates are multiplied by 2.

Did you know? To use Simpson’s Rule, you must have an even number of strips. If \( n \) is odd, the 1-4-2 pattern won't work!

Quick Review Box:
- Midpoint (\( M_n \)): Uses mid-strip heights.
- Trapezium (\( T_n \)): Uses strip-edge heights, 1-2-1 pattern.
- Simpson's (\( S_{2n} \)): Uses strip-edge heights, 1-4-2-4-1 pattern.

5. The MEI "Golden Connections"

In Further Maths B (MEI), a very common task is to find a new estimate using your previous results. You don't always have to start from scratch!

Relationship 1: Improving the Trapezium Rule
If you have \( M_n \) and \( T_n \), you can find the estimate for double the number of strips (\( T_{2n} \)) easily:
\( T_{2n} = \frac{1}{2}(M_n + T_n) \)

Relationship 2: Building Simpson from the others
This is a favorite in exams! Simpson’s Rule is actually a weighted average of the other two:
\( S_{2n} = \frac{1}{3}(2M_n + T_n) \)
Note: Sometimes written as \( S_{2n} = \frac{2M_n + T_n}{3} \).

Key Takeaway: These connections allow you to quickly improve accuracy without recalculating every single \( y \) value.

6. Convergence and Errors

Don't worry if "Order of Convergence" sounds scary; it just means "how fast does the error disappear as we add more strips?"

  • Midpoint and Trapezium Rules: These are second-order methods. If you double the number of strips, the error roughly divides by 4 (\( 2^2 \)).
  • Simpson’s Rule: This is a fourth-order method. If you double the number of strips, the error roughly divides by 16 (\( 2^4 \))! This is why Simpson's is so much more accurate.

Concavity and Over/Under Estimates

Will your estimate be too big or too small? Look at the shape of the graph (the concavity):

  • Concave Up (Happy face \(\cup\)): The Trapezium rule will overestimate (the straight lines sit above the curve). The Midpoint rule will underestimate.
  • Concave Down (Sad face \(\cap\)): The Trapezium rule will underestimate. The Midpoint rule will overestimate.

Key Takeaway: If \( T_n \) is an overestimate and \( M_n \) is an underestimate, the true value is somewhere in the middle!

Summary Checklist

Can you:
1. Calculate strip width \( h \)?
2. Apply the Midpoint, Trapezium, and Simpson's formulas correctly?
3. Remember the 1-4-2-4-1 weighting for Simpson's?
4. Use the "Golden Connections" to find \( T_{2n} \) and \( S_{2n} \)?
5. Determine if an estimate is too high or too low based on the curve's shape?

Keep practicing! Numerical methods are all about being methodical. Double-check your table of values, and you'll do great!