Welcome to Numerical Integration!
In your Pure Mathematics studies, you’ve learned how to integrate many functions exactly. But what happens when you meet a function that is impossible to integrate using standard rules, like \( e^{-x^2} \)?
This is where Numerical Integration comes to the rescue! Instead of looking for an exact algebraic answer, we use clever shapes to estimate the area under a curve. Think of it like measuring the area of a circular rug by covering it with several small rectangular tiles—the more tiles you use, the more accurate your measurement becomes. In this chapter, we will look at three main ways to do this: the Midpoint Rule, the Trapezium Rule, and Simpson’s Rule.
1. The Core Concepts: Strips and Widths
To estimate the area under a curve between \(x = a\) and \(x = b\), we divide the area into \(n\) strips.
The width of each strip is called \(h\).
The formula for the strip width is:
\(h = \frac{b - a}{n}\)
Don't worry if this seems like a lot of letters! Just remember: (End point minus Start point) divided by the number of pieces.
Quick Review: The Key Terms
Ordinates: These are the \(y\)-values of the points on the curve.
Interval: The total distance from \(a\) to \(b\).
Concavity: Whether the curve bends upwards (like a smile) or downwards (like a frown).
2. The Midpoint Rule (\(M_n\))
The Midpoint Rule estimates the area using rectangles. However, instead of using the height at the start or end of the strip, we use the height at the exact middle of each strip.
The Formula:
\(M_n = h(y_{\frac{1}{2}} + y_{\frac{3}{2}} + ... + y_{n-\frac{1}{2}})\)
Note: The \(y\)-values here are the heights at the middle of each strip.
Analogy: Imagine you are building a staircase to fit under a curved roof. If you measure the height at the middle of each step, the part of the step that "sticks out" over the curve roughly cancels out the "gap" under the curve!
3. The Trapezium Rule (\(T_n\))
Instead of flat-topped rectangles, we use trapezia (shapes with slanted tops). We connect the points on the curve with straight lines.
The Formula:
\(T_n = \frac{1}{2}h(y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1}))\)
Memory Trick: Add the first and last heights, then add two of every height in between. Finally, multiply the whole thing by half the strip width.
Common Mistake: Students often forget that for \(n\) strips, there are \(n+1\) heights (\(y\)-values). Always count your \(y\)-values carefully!
Concavity and Errors
The Trapezium Rule isn't perfect. Its accuracy depends on the shape of the curve:
• If the curve is concave upwards (bends like a U), the Trapezium Rule overestimates the area because the straight lines stay above the curve.
• If the curve is concave downwards (bends like an upside-down U), the Trapezium Rule underestimates the area.
Midpoint Rule usually has the opposite error to the Trapezium Rule!
4. Simpson’s Rule (\(S_{2n}\))
This is the "pro" version of numerical integration. Instead of straight lines, it uses parabolas (curves) to fit the top of the strips. This makes it much more accurate!
The Formula:
\(S_{2n} = \frac{1}{3}h(y_0 + y_{2n} + 4(y_1 + y_3 + ... + y_{2n-1}) + 2(y_2 + y_4 + ... + y_{2n-2}))\)
Wait, why \(2n\)? In Simpson’s rule, we group strips in pairs, so we always need an even number of strips.
Memory Aid: "First + Last + 4 times the Odds + 2 times the Evens." Multiply by \(h/3\).
Did you know? Even though it’s named after Thomas Simpson, this method was actually used by mathematicians long before him, including Kepler to calculate the volume of wine barrels!
5. Relationships Between the Methods
One of the coolest parts of the MEI syllabus is seeing how these methods are linked. You can actually calculate more complex estimates using simpler ones!
The "Magic" Connections:
1. \(T_{2n} = \frac{1}{2}(M_n + T_n)\)
This means if you know the Midpoint and Trapezium estimates for \(n\) strips, the Trapezium estimate for double the strips is just their average!
2. \(S_{2n} = \frac{1}{3}(2M_n + T_n)\)
3. \(S_{2n} = \frac{1}{3}(4T_{2n} - T_n)\)
Quick Review: Order of Accuracy
• Midpoint and Trapezium Rules are second-order methods. If you halve the strip width (\(h\)), the error roughly divides by 4.
• Simpson’s Rule is a fourth-order method. It is much more powerful! If you halve \(h\), the error roughly divides by 16.
6. Working with Spreadsheets
In your exam, you might see a screenshot of a spreadsheet. You don't need to be a computer expert, but you should know how the formulas look.
• A cell like B4 refers to the value in Column B, Row 4.
• To find the next \(x\)-value, the formula is usually =A4 + $D$2 (where D2 is the strip width \(h\)).
• The $ sign is very important—it "locks" the cell so it doesn't change when you drag the formula down.
Summary: Key Takeaways
• Numerical Integration is for finding the area when we can't do it algebraically.
• h is the width of one strip: \(h = \frac{b-a}{n}\).
• Concavity tells you if you have over- or under-estimated.
• Simpson's Rule is the most accurate and uses quadratic (parabolic) curves.
• Calculators: Always make sure your calculator is in Radians mode if the function involves Trigonometry!
• Step-by-step: 1. Find \(h\). 2. List your \(x\)-values. 3. Calculate your \(y\)-values. 4. Plug them into the formula provided in the formula booklet.
Don't worry if the formulas look scary! You will always have the formula booklet in the exam. Your main job is to identify the correct \(y\)-values and plug them in carefully.