Integration (HSC SSCE Mathematics Advanced): Revision Notes

The Trapezoidal Rule

Why we need approximation methods

Sometimes we cannot calculate definite integrals exactly using antiderivatives. This happens in two main situations:

In these cases, we need approximation methods to estimate the value of definite integrals.

The basic concept

The trapezoidal rule provides a straightforward way to approximate integrals. Instead of finding the exact area under a curve, we replace the curve with a straight line (called a chord) connecting the two endpoints. This creates a trapezium shape whose area we can easily calculate.

To find the area of the trapezium, we use the formula:

$\text{Area of trapezium} = \text{width} \times \text{average of parallel sides}$

For a function $f(x)$ on the interval $[a, b]$:

• Width = $b - a$
• Average of parallel sides = $\frac{f(a) + f(b)}{2}$

Therefore:

$\text{Area of trapezium} = \frac{b-a}{2}(f(a) + f(b))$

We use this area as our approximation for the integral.

Single subinterval formula

For a continuous function $f(x)$ on the closed interval $[a, b]$:

$\int_a^b f(x)\,dx \approx \frac{b-a}{2}(f(a) + f(b))$

Improving accuracy with multiple subintervals

We can improve the accuracy of our approximation by dividing the interval $[a, b]$ into several smaller subintervals and applying the trapezoidal rule to each one. This gives a better fit to the curve.

Let's see this method applied to the reciprocal function $y = \frac{1}{x}$:

$x$	$1$	$2$	$3$	$4$	$5$
$\frac{1}{x}$	$1$	$\frac{1}{2}$	$\frac{1}{3}$	$\frac{1}{4}$	$\frac{1}{5}$

Understanding concavity and accuracy

The shape of the curve determines whether the trapezoidal rule overestimates or underestimates the integral.

How concavity affects estimates

We can test concavity using the second derivative $\frac{d^2y}{dx^2}$:

• If $\frac{d^2y}{dx^2} > 0$, the curve is concave up
• If $\frac{d^2y}{dx^2} < 0$, the curve is concave down

The general formula for multiple subintervals

When using many subintervals, performing separate calculations becomes tedious. We can develop a single formula that combines all the applications of the trapezoidal rule.

Setting up the division

To divide the interval $[a, b]$ into $n$ equal subintervals, each of width $h$:

$h = \frac{b-a}{n}$

Label the division points as:

$x_0 = a, \quad x_1 = a + h, \quad x_2 = a + 2h, \quad \ldots, \quad x_{n-1} = a + (n-1)h, \quad x_n = b$

In general: $x_r = a + rh$ for $r = 0, 1, 2, \ldots, n-1, n$

Deriving the formula

Applying the trapezoidal rule to each subinterval:

$\int_a^b f(x)\,dx = \int_a^{x_1} f(x)\,dx + \int_{x_1}^{x_2} f(x)\,dx + \cdots + \int_{x_{n-1}}^b f(x)\,dx$

$\approx \frac{h}{2}(f(a) + f(x_1)) + \frac{h}{2}(f(x_1) + f(x_2)) + \cdots + \frac{h}{2}(f(x_{n-1}) + f(b))$

Notice that the middle values appear twice (once as the right endpoint of one trapezium and once as the left endpoint of the next). Collecting like terms:

$\int_a^b f(x)\,dx \approx \frac{h}{2}(f(a) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(b))$

The formula

For a continuous function $f(x)$ on $[a, b]$, using $n$ subintervals:

$\int_a^b f(x)\,dx \approx \frac{h}{2}(f(a) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(b))$

where $h = \frac{b-a}{n}$ and $x_r = a + rh$ for $r = 1, 2, \ldots, n-1$

An alternative form, using nested brackets:

$\int_a^b f(x)\,dx \approx \frac{b-a}{2n}(f(a) + f(b) + 2(f(x_1) + f(x_2) + \cdots + f(x_{n-1})))$

Practical tips for using the formula

Real-world application: experimental data

Often we need to estimate integrals when we only have experimental measurements, without knowing the actual function formula. Here's an example using real hydrological data.

Using technology

The trapezoidal rule formula is well-suited for spreadsheet calculations. While detailed step-by-step instructions vary between software versions, the general approach involves:

• Creating a row of x-values (using formulas to generate equal spacing)
• Calculating corresponding function values
• Applying the coefficient pattern (1 for endpoints, 2 for middle terms)
• Summing and multiplying by $\frac{h}{2}$

This allows you to easily experiment with different numbers of subintervals to see how accuracy improves.