Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.8
📋 Learning Intentions
- State the trapezoidal rule formula and identify each component, including the strip width \(h\) and the function values \(y_0, y_1, \ldots, y_n\).
- Apply the trapezoidal rule to approximate a definite integral using a given number of subintervals, both by hand and with the GDC.
- Calculate the strip width \(h = \frac{b - a}{n}\) and construct a table of \(x\)- and \(y\)-values before substituting into the formula.
- Interpret the result of the trapezoidal rule as an approximation to the area under a curve, and comment on whether the approximation is an overestimate or underestimate.
- Determine how increasing the number of strips improves the accuracy of the approximation.
Real-World Introduction
A hydrologist needs to find the total volume of water that flowed through a river channel during a flood event. Flow rate data is recorded every hour, but the rate changes continuously and cannot be described by a neat algebraic formula. The only available information is a table of measurements at discrete time intervals.
In situations like this, exact integration is impossible — there is no function to integrate. Instead, mathematicians approximate the area under the data curve by dividing it into strips and treating each strip as a trapezoid. This is the core idea behind the trapezoidal rule, one of the most widely used numerical integration techniques in science and engineering.
🌊 Why This Matters The trapezoidal rule is used in environmental science to estimate pollutant exposure from concentration-time graphs, in pharmacokinetics to calculate the area under a drug concentration curve (AUC), and in engineering to find work done from force-displacement data — all situations where no algebraic formula exists for the function.
Core Concepts & Definitions
When a definite integral \(\int_a^b f(x)\,dx\) cannot be evaluated exactly — either because the function has no elementary antiderivative, or because only data values are available — a numerical method is needed. The simplest and most examinable such method in Math AI is the trapezoidal rule.
📐 Key Definition — The Trapezoidal Rule The trapezoidal rule approximates the area