Integration, definite integrals and areas

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 6 min read · Updated: 7 Sept 2026

Introduction to Integration

Welcome to the world of Integration! If differentiation is the process of finding the "gradient" or the rate of change, integration is essentially the opposite: it's the process of "undoing" a derivative to find the original function. Think of it like being a detective—if differentiation tells you how fast a car is moving, integration helps you figure out how far it has traveled.

In this chapter, we will explore indefinite integrals (finding general functions), definite integrals (finding specific values), and how these tools allow us to calculate the area of complex shapes that don't have standard geometric formulas.

1. Indefinite Integration: The "Anti-Derivative"

Integration is often called anti-differentiation. If we have a derivative \(f'(x)\), we integrate it to find the original function \(f(x)\).

The Power Rule for Integration

For any term \(ax^n\), where \(n\) is a rational number and \(n \neq -1\), the rule is: \( \int ax^n dx = \frac{ax^{n+1}}{n+1} + C \)

Step-by-Step:
1. Add 1 to the power: \(n \to n+1\)
2. Divide the term by the new power.
3. Always add the constant of integration, \(C\).

Why the \(C\)? When we differentiate a constant (like 5 or 100), it becomes zero. When we go backward, we don't know what that original constant was, so we use \(C\) to represent any possible number.

Integration of Standard Functions (SL & HL)

You need to be familiar with these standard integrals:

\( \int e^x dx = e^x + C \)
\( \int \cos x dx = \sin x + C \)
\( \int \sin x dx = -\cos x + C \)
\( \int \frac{1}{x} dx = \ln|x| + C \) (This is the case where \(n = -1\))

Boundary Conditions

If you are given a specific point \((x, y)\) that the original curve passes through, you can solve for \(C\). Example: If \(f'(x) = 2x\) and the curve passes through \((1, 5)\).
Integrating gives \(f(x) = x^2 + C\).
Substitute the point: \(5 = (1)^2 + C \implies C = 4\).
The specific function is \(f(x) = x^2 + 4\).

Key Takeaway: Indefinite integration gives a family of curves. To find the specific curve, you need a starting point (boundary condition).

2. Integration Techniques

Linear Composites (The \(ax+b\) Rule)

If you are integrating a function where the "inside" is a linear expression like \(ax+b\), you integrate as normal and then divide by \(a\) (the coefficient of \(x\)).

\( \int f(ax+b) dx = \frac{1}{a} F(ax+b) + C \)

Example: \( \int \cos(3x+2) dx = \frac{1}{3}\sin(3x+2) + C \)

Integration by Substitution (Reverse Chain Rule)

This is used for expressions in the form \( \int k g'(x) f(g(x)) dx \). We look for a part of the function whose derivative is also present in the integral.

Steps:
1. Let \(u = g(x)\).
2. Find \( \frac{du}{dx} = g'(x) \), which means \( dx = \frac{du}{g'(x)} \).
3. Substitute \(u\) and \(du\) into the integral to simplify it.
4. Integrate with respect to \(u\).
5. Substitute \(g(x)\) back in for \(u\).

Quick Tip: If you see a function and its derivative multiplied together, substitution is almost always the right path!

3. Definite Integrals

A definite integral has upper and lower limits (boundaries) and results in a numerical value rather than a function with \(C\).

The Fundamental Theorem of Calculus states:
\( \int_{a}^{b} f(x) dx = [F(x)]_a^b = F(b) - F(a) \)

where \(F(x)\) is the integral of \(f(x)\).

Using Technology: In Paper 2, you are expected to use your GDC (Graphic Display Calculator) to evaluate definite integrals. Make sure you know how to find the integral tool in your calculator's math menu!

4. Finding Areas

Integration is a powerful tool for finding the area "under" a curve.

Area between a Curve and the x-axis

The area between \(y = f(x)\) and the x-axis from \(x=a\) to \(x=b\) is given by \( \int_a^b f(x) dx \).

Important Note on Negative Areas:
If the curve is above the x-axis, the integral is positive.
If the curve is below the x-axis, the integral will be negative. Area must always be positive, so if a region is below the axis, take the absolute value or calculate \( -\int f(x) dx \).

Common Mistake: If you integrate a curve that crosses the x-axis (partly above, partly below) in one go, the negative part will cancel out the positive part! To find the total area, you must calculate the area of each section separately and add their absolute values.

Area Between Two Curves

To find the area between curve \(f(x)\) and curve \(g(x)\) from \(x=a\) to \(x=b\):
\( \text{Area} = \int_a^b |f(x) - g(x)| dx \)

In simple terms: (Top Curve) minus (Bottom Curve).

Key Takeaway: Always sketch the functions first (or use your GDC) to see which curve is "on top" and where they intersect.

5. Higher Level (AHL) Extensions

This section is for HL students only.

Additional Standard Integrals

HL students must also be able to integrate functions like:
\( \int \tan x dx \), \( \int a^x dx \), and \( \int \frac{1}{x^2+a^2} dx \) (which involves \(\arctan\)).
Always check your Formula Booklet during the exam for these complex forms!

Integration by Parts

This is used for the product of two functions that cannot be solved by substitution. It is the "reverse product rule":
\( \int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx \)

The LIATE Rule: When choosing which part should be \(u\), follow this priority order: Logarithmic, Inverse Trig, Algebraic (\(x^2\)), Trigonometric, Exponential.

Area with the y-axis

Sometimes it is easier to find the area between a curve and the y-axis. To do this, rearrange the function to \(x = f(y)\) and integrate with respect to \(y\):
\( \text{Area} = \int_c^d x dy \)

Volumes of Revolution

If you rotate a 2D area around an axis, you create a 3D solid. We can calculate its volume!

Rotation around the x-axis: \( V = \int_a^b \pi y^2 dx \)
Rotation around the y-axis: \( V = \int_c^d \pi x^2 dy \)

Memory Aid: It looks like the area formula, but you square the function and multiply by \(\pi\) (because the cross-sections are circles).

Summary Checklist

Can I use the power rule and include the \(+C\)?
Do I know how to find \(C\) using a given point?
Can I integrate \(\sin x\), \(\cos x\), and \(e^x\)?
Do I remember to divide by \(a\) for linear composites?
Can I find the area between a curve and the x-axis, taking care of regions below the axis?
HL: Am I comfortable with Integration by Parts and Volumes of Revolution?

Quick check

Can you answer these now?

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

What is integration in the context of calculus?

Integration is the process of "undoing" a derivative to find the original function, often called anti-differentiation.

What is the Power Rule for integration?

\( \int ax^n dx = \frac{ax^{n+1}}{n+1} + C \), provided that \( n \neq -1 \).

Why must you always include the constant of integration \( C \) in indefinite integrals?

The derivative of any constant is zero, so when we integrate, we must account for any possible constant that might have been lost during differentiation.

What is the integral of \( \frac{1}{x} \)?

\( \int \frac{1}{x} dx = \ln|x| + C \).

How do you solve for the constant of integration \( C \)?

Substitute the coordinates of a known point \( (x, y) \) into the integrated function and solve the resulting equation for \( C \).

How do you integrate a linear composite function of the form \( f(ax+b) \)?

Integrate the function as normal, then divide the entire result by the coefficient of \( x \) (which is \( a \)): \( \frac{1}{a} F(ax+b) + C \).

When should you use the Integration by Substitution method?

Use substitution when the integrand contains a composite function where the derivative of the "inside" part is also present as a factor.

How do you evaluate a definite integral \( \int_{a}^{b} f(x) dx \)?

Find the integral \( F(x) \), then calculate \( F(b) - F(a) \).

How do you calculate the total area when a curve crosses the x-axis?

Calculate the area of the sections above and below the x-axis separately, then add their absolute values together.

What is the formula for finding the area between two curves \( f(x) \) and \( g(x) \)?

\( \text{Area} = \int_a^b (\text{top curve} - \text{bottom curve}) dx \).

What is the Integration by Parts formula used by HL students?

\( \int u \frac{dv}{dx} dx = uv - ∫ v \frac{du}{dx} dx \).

What are the formulas for Volumes of Revolution around the x and y axes?

Around x-axis: \( V = \int_a^b \pi y^2 dx \); Around y-axis: \( V = \int_c^d \pi x^2 dy \).

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