QCE Mathematical Methods - Unit 4 - Further integration

Definite Integrals and the Fundamental Theorem | QCE Mathematical Methods

Learn QCE Mathematical Methods definite integrals, area as a limit of sums and the fundamental theorem of calculus.

Updated 2026-05-18 - 4 min read

An indefinite integral gives a family of anti-derivatives. A definite integral gives a number. In Methods, that number often represents area, total change, displacement or accumulated quantity.

Area as a limit of sums

To estimate area under $y=f(x)$ from $x=a$ to $x=b$, you can split the interval into thin strips. Each strip has approximate area:

$ f(x_i)\delta x_i $

Adding the strips gives:

$ \sum_i f(x_i)\delta x_i $

As the strips become thinner, this sum approaches the definite integral:

$ \int_a^b f(x)\,dx $

If the interval is split into $n$ equal subintervals, the width is:

$ \Delta x=\frac{b-a}{n}. $

With left endpoints:

$ \int_a^b f(x)\,dx\approx \sum_{i=0}^{n-1}f(x_i)\Delta x. $

With right endpoints:

$ \int_a^b f(x)\,dx\approx \sum_{i=1}^{n}f(x_i)\Delta x. $

For an increasing positive function, left rectangles usually underestimate and right rectangles usually overestimate. For a decreasing positive function, the reverse happens. This is why a quick sketch matters before judging an approximation.

As $n$ becomes very large, $\Delta x$ tends toward $0$, and the approximation approaches the exact definite integral:

$ \int_a^b f(x)\,dx=\lim_{\Delta x\to0}\sum_i f(x_i)\Delta x_i. $

The notation $\Delta x_i$ allows unequal subinterval widths, although most Methods calculations use equal widths.

Estimating before exact integration

Approximation is not just a backup when you cannot integrate. It helps you understand what the integral represents. If $f(x)$ is a velocity, then each term $f(x_i)\Delta x$ is approximately:

$ \text{velocity}\times\text{time} $

which is a small displacement. Adding the pieces gives total displacement. If $f(x)$ is a marginal cost, each term is approximately extra cost over a small production interval, and the integral gives total change in cost.

For equal-width rectangles, always identify:

• the interval $[a,b]$
• the number of strips $n$
• the width $\Delta x=\frac{b-a}{n}$
• whether the heights use left endpoints, right endpoints or another chosen point

This structure prevents the most common Riemann-sum mistake: using the wrong $x$ values for the rectangle heights.

The fundamental theorem of calculus

If $F'(x)=f(x)$, then:

$ \int_a^b f(x)\,dx=F(b)-F(a) $

This is what makes exact calculation possible. Instead of adding infinitely many thin rectangles, find an anti-derivative and subtract its endpoint values.

The integrand must be continuous on the interval for the basic Methods version of the theorem to apply cleanly. For example, do not apply the theorem across a vertical asymptote without splitting the interval and thinking about whether the integral exists.

For definite integrals, the constant of integration cancels:

$ [F(x)+c]_a^b=(F(b)+c)-(F(a)+c)=F(b)-F(a). $

That is why you do not write $+c$ in the final value of a definite integral.

The bracket notation is a compact way to show the substitution:

$ [F(x)]_a^b=F(b)-F(a) $

Read it as "upper endpoint minus lower endpoint". Reversing the order changes the sign:

$ \int_b^a f(x)\,dx=-\int_a^b f(x)\,dx $

and if the bounds are the same:

$ \int_a^a f(x)\,dx=0 $

because there is no interval width to accumulate over.

Worked example

Worked rectangle estimate

Area and sign

If $f(x)>0$ over the whole interval, the definite integral is the area under the curve. If $f(x)<0$, the definite integral is negative because the curve is below the $x$-axis.

That difference matters. "Evaluate the integral" and "find the area" are not always the same instruction.

For example:

$ \int_0^{2\pi}\sin x\,dx=0, $

because the positive area from $0$ to $\pi$ cancels the negative signed area from $\pi$ to $2\pi$. The total area is:

$ \int_0^\pi\sin x\,dx-\int_\pi^{2\pi}\sin x\,dx. $

So a zero definite integral does not always mean no area exists. It can mean signed regions cancelled.

Quick check

Sources

• QCAA Mathematical Methods subject page
• QCAA Mathematical Methods 2025 syllabus
• OpenStax Calculus Volume 1: The Definite Integral
• OpenStax Calculus Volume 1: The Fundamental Theorem of Calculus