Definite integration and its applications

Q: What does the definite integral \(\int_{a}^{b} f(x) \,dx\) represent geometrically when \(f(x) \ge 0\)?

It represents the exact area bounded by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\).

Q: How does the Fundamental Theorem of Calculus (FTC) evaluate a definite integral \(\int_{a}^{b} f(x) \,dx\)?

\(\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(b) - F(a)\), where \(F(x)\) is an antiderivative of \(f(x)\) such that \(F'(x) = f(x)\).

Q: What are the key algebraic properties of definite integrals regarding reversed limits and identical limits?

Reversing limits negates the value: \(\int_{b}^{a} f(x) \,dx = -\int_{a}^{b} f(x) \,dx\); identical limits yield zero: \(\int_{a}^{a} f(x) \,dx = 0\).

Q: What essential step must be performed when evaluating a definite integral using the method of substitution?

You must transform the upper and lower limits of integration from the original variable \(x\) to the new substitution variable \(u\).

Q: How do you find the area bounded between a curve \(y = f(x)\) and the \(x\)-axis over an interval \([a, b]\) if the curve dips below the \(x\)-axis?

Split the integral at the \(x\)-intercepts and integrate the absolute value: \(\text{Area} = \int_{a}^{b} |f(x)| \,dx\), negating parts where \(f(x)

Q: How is definite integration used to find the total change of a quantity from its rate of change \(R(t) = \frac{dQ}{dt}\)?

The total accumulation of change from \(t_1\) to \(t_2\) is \(\Delta Q = Q(t_2) - Q(t_1) = \int_{t_1}^{t_2} R(t) \,dt\).

Q: What is the Trapezoidal Rule formula for approximating \(\int_{a}^{b} f(x) \,dx\) with \(n\) equal sub-intervals of width \(h\)?

\(\int_{a}^{b} f(x) \,dx \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + \dots + y_{n-1}) + y_n]\), where \(h = \frac{b - a}{n}\).

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Mathematics M1 (Calculus and Statistics) · Senior Secondary (HKDSE) · 6 min read · Updated: 24 Aug 2026

Definite Integration and Its Applications

Hey everyone! Welcome to the exciting world of Definite Integration. If you've ever wondered how we can find the exact area of curved shapes or calculate the total accumulation of change, you've come to the right place. We are going to break it down step-by-step.

In this chapter, we'll learn how to calculate exact areas, see how definite integration is connected to differentiation via the Fundamental Theorem of Calculus, explore the Trapezoidal Rule for approximation, and solve real-world problems.

1. What is Definite Integration? The Big Idea

The Area Problem

Finding the area of a rectangle (length × width) or a triangle (½ × base × height) is easy. But what about the area under a curve? There is no single elementary geometric formula for that!

Slicing It Up: The Rectangle Idea

The core idea is simple: let's approximate the area using something we know – rectangles! We can slice the area under the curve into lots of thin vertical rectangles and add up their areas.

The more rectangles we use, and the thinner we make them, the closer our approximation gets to the true area. Definite integration represents the limit of this sum as the number of rectangles approaches infinity and their width approaches zero.

The Notation for Definite Integration

This process is written with a special notation:

\(\int_{a}^{b} f(x) \,dx\)

The Integral Sign (\(\int\)): An elongated 'S' standing for 'Sum'.
The Limits of Integration (\(a\) and \(b\)): \(a\) is the lower limit and \(b\) is the upper limit on the \(x\)-axis.
The Integrand (\(f(x)\)): The height of the curve at each point \(x\).
\(dx\): The infinitesimal width of each rectangle.

Key Takeaway: When \(f(x) \ge 0\), the definite integral \(\int_{a}^{b} f(x) \,dx\) represents the exact area bounded by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\).

2. The Magic Tool: The Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus (FTC) connects integration with differentiation. It provides a simple method to evaluate definite integrals algebraically:

\(\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(b) - F(a)\)

where \(F(x)\) is an antiderivative of \(f(x)\), meaning \(F'(x) = f(x)\).

Step-by-Step Evaluation

Let's evaluate \(\int_{1}^{3} 2x \,dx\):

Find the antiderivative: The antiderivative of \(2x\) is \(x^2\) (the constant \(+ C\) cancels out in definite integration: \((b^2+C) - (a^2+C) = b^2 - a^2\)).
Substitute the upper limit (\(b = 3\)): \(F(3) = 3^2 = 9\).
Substitute the lower limit (\(a = 1\)): \(F(1) = 1^2 = 1\).
Subtract: \(F(3) - F(1) = 9 - 1 = 8\).

Common Mistake to Avoid

Always subtract the lower limit result from the upper limit result: \(F(b) - F(a)\), never \(F(a) - F(b)\).

3. Properties of Definite Integrals

1. Reversing the Limits

\(\int_{b}^{a} f(x) \,dx = - \int_{a}^{b} f(x) \,dx\)
2. Identical Limits

\(\int_{a}^{a} f(x) \,dx = 0\)
3. Splitting the Interval

\(\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx\)
4. Constant Multiples

\(\int_{a}^{b} k f(x) \,dx = k \int_{a}^{b} f(x) \,dx\)
5. Sum and Difference

\(\int_{a}^{b} [f(x) \pm g(x)] \,dx = \int_{a}^{b} f(x) \,dx \pm \int_{a}^{b} g(x) \,dx\)

Dummy Variable

The variable of integration is a dummy variable: \(\int_{a}^{b} f(x) \,dx = \int_{a}^{b} f(t) \,dt = \int_{a}^{b} f(u) \,du\).

4. Methods of Integration: Substitution with Limits

When using integration by substitution for definite integrals, always transform the limits of integration from \(x\) to \(u\).

Example: Evaluate \(\int_{0}^{2} 2x(x^2 + 1)^3 \,dx\)

Let \(u = x^2 + 1\), then \(du = 2x \,dx\).
Change the limits: When \(x = 0\), \(u = 0^2 + 1 = 1\). When \(x = 2\), \(u = 2^2 + 1 = 5\).
Rewrite and integrate: \(\int_{1}^{5} u^3 \,du = \left[ \frac{u^4}{4} \right]_{1}^{5} = \frac{5^4}{4} - \frac{1^4}{4} = \frac{625 - 1}{4} = 156\).

5. Applications of Definite Integration

Finding Areas of Plane Figures

1. Area between a curve and the \(x\)-axis:

If \(f(x) \ge 0\) on \([a, b]\): \(\text{Area} = \int_{a}^{b} f(x) \,dx\).
If \(f(x) \le 0\) on \([a, b]\): \(\text{Area} = -\int_{a}^{b} f(x) \,dx = \left| \int_{a}^{b} f(x) \,dx \right|\).
If the curve crosses the \(x\)-axis at \(x = c\), split the integral: \(\text{Area} = \int_{a}^{c} |f(x)| \,dx + \int_{c}^{b} |f(x)| \,dx\).

2. Area between two curves:

If \(f(x) \ge g(x)\) on the interval \([a, b]\), the area enclosed between \(y = f(x)\) and \(y = g(x)\) from \(x = a\) to \(x = b\) is:

\(\text{Area} = \int_{a}^{b} [f(x) - g(x)] \,dx = \int_{a}^{b} (\text{upper curve} - \text{lower curve}) \,dx\)

Example: Find the area enclosed between \(y = 4 - x^2\) and \(y = x + 2\).

Find intersection points by setting \(4 - x^2 = x + 2 \implies x^2 + x - 2 = 0 \implies (x + 2)(x - 1) = 0\), so \(x = -2\) and \(x = 1\).
On \([-2, 1]\), \(4 - x^2 \ge x + 2\).
\(\text{Area} = \int_{-2}^{1} [(4 - x^2) - (x + 2)] \,dx = \int_{-2}^{1} (2 - x - x^2) \,dx = \left[ 2x - \frac{x^2}{2} - \frac{x^3}{3} \right]_{-2}^{1} = \left( 2 - \frac{1}{2} - \frac{1}{3} \right) - \left( -4 - 2 + \frac{8}{3} \right) = \frac{7}{6} - \left(-\frac{10}{3}\right) = \frac{9}{2} = 4.5\).

Rates of Change and Total Accumulation

If \(R(t) = \frac{dQ}{dt}\) is the rate of change of a quantity \(Q\) with respect to time \(t\), then the total change in \(Q\) from \(t = t_1\) to \(t = t_2\) is given by the definite integral:

\(\Delta Q = Q(t_2) - Q(t_1) = \int_{t_1}^{t_2} R(t) \,dt\)

6. Numerical Integration: The Trapezoidal Rule

When a function is difficult or impossible to integrate analytically, we can approximate the definite integral using the Trapezoidal Rule.

Formula

Divide the interval \([a, b]\) into \(n\) sub-intervals of equal width \(h = \frac{b - a}{n}\), with partition points \(x_0 = a, x_1, x_2, \dots, x_n = b\) and corresponding values \(y_0, y_1, y_2, \dots, y_n\) where \(y_i = f(x_i)\):

\(\int_{a}^{b} f(x) \,dx \approx \frac{h}{2} \left[ y_0 + 2(y_1 + y_2 + \dots + y_{n-1}) + y_n \right] = \frac{h}{2} \left[ (\text{first} + \text{last}) + 2(\text{sum of middle terms}) \right]\)

Over-estimation vs Under-estimation

If the curve is concave upwards (\(f''(x) > 0\)) on \([a, b]\), the trapezoid chords lie above the curve, so the trapezoidal rule gives an over-estimate.
If the curve is concave downwards (\(f''(x) < 0\)) on \([a, b]\), the trapezoid chords lie below the curve, so the trapezoidal rule gives an under-estimate.

7. Chapter Summary & Quick Review

Fundamental Theorem of Calculus: \(\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(b) - F(a)\).
Substitution: Always change the limits of integration into the new variable \(u\).
Area between curves: \(\text{Area} = \int_{a}^{b} [f(x) - g(x)] \,dx\) where \(f(x) \ge g(x)\).
Total Change: \(\int_{t_1}^{t_2} R(t) \,dt = Q(t_2) - Q(t_1)\).
Trapezoidal Rule: \(\int_{a}^{b} f(x) \,dx \approx \frac{h}{2}[y_0 + 2(y_1 + \dots + y_{n-1}) + y_n]\), where \(h = \frac{b-a}{n}\).

References

Mathematics Education - Curriculum Documents — Education Bureau (EDB), Hong Kong SAR Government

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does the definite integral \(\int_{a}^{b} f(x) \,dx\) represent geometrically when \(f(x) \ge 0\)?

It represents the exact area bounded by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\).

How does the Fundamental Theorem of Calculus (FTC) evaluate a definite integral \(\int_{a}^{b} f(x) \,dx\)?

\(\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(b) - F(a)\), where \(F(x)\) is an antiderivative of \(f(x)\) such that \(F'(x) = f(x)\).

What are the key algebraic properties of definite integrals regarding reversed limits and identical limits?

Reversing limits negates the value: \(\int_{b}^{a} f(x) \,dx = -\int_{a}^{b} f(x) \,dx\); identical limits yield zero: \(\int_{a}^{a} f(x) \,dx = 0\).

What essential step must be performed when evaluating a definite integral using the method of substitution?

You must transform the upper and lower limits of integration from the original variable \(x\) to the new substitution variable \(u\).

How do you find the area bounded between a curve \(y = f(x)\) and the \(x\)-axis over an interval \([a, b]\) if the curve dips below the \(x\)-axis?

Split the integral at the \(x\)-intercepts and integrate the absolute value: \(\text{Area} = \int_{a}^{b} |f(x)| \,dx\), negating parts where \(f(x) < 0\).

What is the formula for the area enclosed between two curves \(y = f(x)\) and \(y = g(x)\) from \(x = a\) to \(x = b\) where \(f(x) \ge g(x)\)?

\(\text{Area} = \int_{a}^{b} [f(x) - g(x)] \,dx = \int_{a}^{b} (\text{upper curve} - \text{lower curve}) \,dx\).

How is definite integration used to find the total change of a quantity from its rate of change \(R(t) = \frac{dQ}{dt}\)?

The total accumulation of change from \(t_1\) to \(t_2\) is \(\Delta Q = Q(t_2) - Q(t_1) = \int_{t_1}^{t_2} R(t) \,dt\).

What is the Trapezoidal Rule formula for approximating \(\int_{a}^{b} f(x) \,dx\) with \(n\) equal sub-intervals of width \(h\)?

\(\int_{a}^{b} f(x) \,dx \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + \dots + y_{n-1}) + y_n]\), where \(h = \frac{b - a}{n}\).

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