Definite integrals and area under curves

IB Mathematics Applications and Interpretation HL· Topic 5: Calculus, 5.8 Definite integrals and area· 15 min read

1. Definite Integrals: Definition and Evaluation★★☆☆☆⏱ 5 min

📘 Definition

Definite Integral

\int_a^b f(x) \, dx

The definite integral of a function f(x) from lower limit a to upper limit b is computed via the Fundamental Theorem of Calculus, producing a numerical result rather than a function.

Example:

\int_2^4 2x , dx = [x^2]_2^4 = 16 - 4 = 12

The Fundamental Theorem of Calculus (FTC) Part 2 tells us that if $F (x)$ is an antiderivative of $f (x)$ , meaning $F^{'} (x) = f (x)$ , then:

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

📐 Worked Example

Evaluate the definite integral $\int_{1}^{3} (3 x^{2} + 2 x) d x$

1
First find the general antiderivative of the integrand. The constant of integration cancels out in definite integrals, so we can omit it.
2
$\int (3x^2 + 2x) dx = x^3 + x^2$
3
Write the antiderivative in square brackets with the upper and lower bounds marked:
4
$[x^3 + x^2]_1^3$
5
Evaluate at the upper bound, subtract the value at the lower bound:
6
$(3^3 + 3^2) - (1^3 + 1^2) = (27 + 9) - (1 + 1) = 34$

2. Net Area vs Total Area★★★☆☆⏱ 4 min

A common source of exam confusion is the difference between the value of a definite integral (net area) and the total geometric area between a curve and the x-axis.

📘 Definition

Net Area

The value of the definite integral counts area above the x-axis as positive and area below the x-axis as negative, resulting in a net signed area.

📘 Definition

Total Area

Total geometric area is the sum of the absolute values of each signed region, so all area counts as positive regardless of position relative to the x-axis.

✓ Quick check

Test your understanding of the difference between net and total area:

What is the value of $\int_{- 2}^{2} x d x$ (net area)?
- 0
- 2
- 4
- -4
Reveal answer
0 —
Correct! The area above the axis from 0 to 2 cancels the negative area below from -2 to 0, so net area is 0, while total geometric area is 4.

3. Calculating Total Area Between a Curve and the x-axis★★★☆☆⏱ 6 min

To find the total geometric area between $y = f (x)$ , the x-axis, $x = a$ and $x = b$ , follow this 3-step process:

Find all x-intercepts of $f (x)$ between $a$ and $b$
Split the interval into sub-intervals between consecutive intercepts
Take the absolute value of each sub-integral and add them together

📐 Worked Example

Find the total geometric area between $y = x$ and the x-axis from $x = - 2$ to $x = 2$ .

1
Find the x-intercept to split the interval: $y = x$ crosses the x-axis at $x = 0$ , so we split into $[- 2, 0]$ and $[0, 2]$ .
2
Evaluate the integral over each sub-interval:
3
$\int_{-2}^0 x dx = \left[\frac{1}{2}x^2\right]_{-2}^0 = 0 - 2 = -2$
4
$\int_0^2 x dx = \left[\frac{1}{2}x^2\right]_0^2 = 2 - 0 = 2$
5
Take absolute values of each result and add for total area:
6
$|-2| + |2| = 2 + 2 = 4$

4. Real-World Applications★★★☆☆⏱ 5 min

Definite integrals and area calculations are very common in real-world modelling problems in IB AI HL. For example, the net area under a velocity-time graph gives total displacement, while total area gives total distance travelled.

📐 Worked Example

A ball is thrown straight up, with velocity $v (t) = 20 - 10 t$ m/s, where $t$ is time in seconds after release. Find the total distance travelled by the ball between $t = 0$ and $t = 3$ seconds.

1
Find where velocity changes sign: set $v (t) = 0$ → $20 - 10 t = 0$ → $t = 2$ seconds. Split into $[0, 2]$ (upward movement, positive velocity) and $[2, 3]$ (downward movement, negative velocity).
2
Integrate over each interval:
3
$\int_0^2 (20 - 10t) dt = [20t - 5t^2]_0^2 = 20$
4
$\int_2^3 (20 - 10t) dt = [20t - 5t^2]_2^3 = -5$
5
Add absolute values for total distance:
6
$|20| + |-5| = 25 \text{ meters}$

5. Common Pitfalls

Wrong move:

Forgetting to split the integral at x-intercepts when calculating total area

Why:

Integrating over the entire interval lets negative area cancel positive area, resulting in net area instead of total area

Correct move:

Always find all x-intercepts in the interval, split the integral into sub-regions, then add absolute values of each part

Wrong move:

Assuming the value of the definite integral is always equal to geometric area

Why:

The definite integral gives signed net area, which can be zero or negative even when total area is positive

Correct move:

Check the question wording: if it asks for area, it means total positive geometric area

Wrong move:

Evaluating $F (a) - F (b)$ instead of $F (b) - F (a)$ for the definite integral

Why:

Swapping the bounds flips the sign of the result, leading to an incorrect answer, often negative area

Correct move:

Always subtract the antiderivative evaluated at the lower limit from the value at the upper limit

Wrong move:

Forgetting to take the absolute value of negative sub-integrals for total area

Why:

Leaving negative area as negative reduces the final total, leading to an incorrect result

Correct move:

Take the absolute value of each sub-integral before adding to get total area

6. Quick Reference Cheatsheet

Concept	Formula / Method
Definite integral (FTC)	$\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)$
Net area (integral value)	Area above x-axis minus area below
Total geometric area	Sum of $\int_{x_{i}}^{x_{i + 1}} f (x) d x$ for each interval between intercepts
Area above x-axis	$\int_{a}^{b} f (x) d x, f (x) \geq 0$
Area below x-axis	$- \int_{a}^{b} f (x) d x, f (x) \leq 0$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 2
Area under velocity-time graph
2022 · 1
Evaluate definite integral
2023 · 2
Total area of crossing curve

Going deeper

syllabusIB AI HL Official SyllabusFull topic learning outcomes

What's Next

Now that you have mastered definite integrals and area under curves, you have built a core foundational skill for all advanced integration applications in IB AI HL. Definite integrals are used to calculate volumes of revolution, the average value of a function, and accumulated quantities like population growth or energy use in modelling problems. Understanding the difference between net and total area is also critical for interpreting integration results when working with real-world data and functions in your internal assessment and exams. You can now extend your skills to more complex integration scenarios that appear regularly in IB assessments.