Edexcel IAL - Pure Maths 2- 8.3 Area Approximation Using the Trapezium Rule- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 8.3 Area Approximation Using the Trapezium Rule -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 8.3 Area Approximation Using the Trapezium Rule -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 8.3 Area Approximation Using the Trapezium Rule
Approximation of Area Under a Curve: Trapezium Rule
When a definite integral cannot be evaluated exactly, or when only approximate values are required, the trapezium rule can be used to estimate the area under a curve.
The trapezium rule approximates the region under a curve by dividing it into a number of trapezia.
The Trapezium Rule Formula
To approximate:
\( \displaystyle \int_a^b f(x)\,dx \)
Divide the interval \( [a,b] \) into \( n \) equal strips, where:
\( h = \dfrac{b-a}{n} \)
The trapezium rule gives:
\( \displaystyle \int_a^b f(x)\,dx \approx \dfrac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \)
where \( x_0 = a \) and \( x_n = b \).
Geometric Interpretation
- The curve is replaced by straight line segments.
- Each strip forms a trapezium.
- The sum of trapezium areas approximates the true area.
Accuracy of the Trapezium Rule
- Increasing the number of trapezia improves accuracy.
- If the curve is concave up, the trapezium rule overestimates the area.
- If the curve is concave down, it underestimates the area.
- The error decreases as \( n \) increases.
General Method
| Step | Action |
| 1 | Find the strip width \( h \) |
| 2 | Calculate the x-values |
| 3 | Evaluate \( f(x) \) at each point |
| 4 | Apply the trapezium rule formula |
Example
Use the trapezium rule with 2 strips to approximate:
\( \displaystyle \int_0^2 x^2\,dx \)
▶️ Answer / Explanation
\( h = \dfrac{2-0}{2} = 1 \)
x-values: \( 0,\ 1,\ 2 \)
Function values:
\( f(0)=0,\ f(1)=1,\ f(2)=4 \)
Apply trapezium rule:
\( \dfrac{1}{2}[0 + 2(1) + 4] = \dfrac{1}{2}(6) = 3 \)
Approximate area = 3
Example
Use the trapezium rule with 4 strips to approximate:
\( \displaystyle \int_0^1 \sqrt{2x + 1}\,dx \)
▶️ Answer / Explanation
\( h = \dfrac{1-0}{4} = 0.25 \)
x-values:
\( 0,\ 0.25,\ 0.5,\ 0.75,\ 1 \)
Function values:
\( f(0)=1 \)
\( f(0.25)=\sqrt{1.5} \approx 1.225 \)
\( f(0.5)=\sqrt{2} \approx 1.414 \)
\( f(0.75)=\sqrt{2.5} \approx 1.581 \)
\( f(1)=\sqrt{3} \approx 1.732 \)
Apply trapezium rule:
\( \dfrac{0.25}{2}[1 + 2(1.225 + 1.414 + 1.581) + 1.732] \)
\( = 0.125[1 + 2(4.220) + 1.732] \)
\( = 0.125(11.172) \approx 1.396 \)
Approximate value = 1.396
Example
Explain how the accuracy of the trapezium rule approximation can be improved.
▶️ Answer / Explanation
The accuracy can be improved by:
- Increasing the number of trapezia
- Reducing the strip width \( h \)
- Making the straight-line approximation closer to the curve
As \( n \) increases, the error decreases.