Integration is the exact mathematical reverse of differentiation. If differentiating y = x^3 gives dy/dx = 3x^2, then integrating 3x^2 gives back x^3 + C. The '+ C' is a constant of integration that accounts for any vertical shift.

O-Level Add Maths: Definite Integrals and Area Under Curves

Q: What is the Reverse Power Rule?

To integrate ax^n, you add 1 to the power and divide by the new power: integral of ax^n dx = ax^(n+1) / (n+1) + C. Example: integral of 6x^2 dx = 6x^3/3 + C = 2x^3 + C.

Q: What is a Definite Integral?

A definite integral has upper and lower limits (like integral from 1 to 4). Instead of adding '+ C', you substitute the upper limit into the integrated expression, then subtract the result of substituting the lower limit. The answer is a specific number representing the exact area.

Q: What happens when the curve goes below the x-axis?

The integral gives a NEGATIVE value for areas below the x-axis. You must calculate the areas above and below separately, then add their absolute values. If you integrate straight across, the negative area cancels out the positive area, giving a wrong answer.

A blackboard with integral notation and shaded area diagrams.

How do I find the area under a curve between x=1 and x=4?

Step 1: Integrate the curve equation using the reverse power rule (add 1 to the power, divide by new power). Step 2: Substitute the upper limit (x=4) into the result. Step 3: Subtract the result of substituting the lower limit (x=1). The final number is the exact area in square units. Critical: if the curve dips below the x-axis, you must split the integral at the x-intercept and take the absolute value of the negative part.

1. The Reverse Power Rule
2. Definite vs Indefinite Integrals
3. Area Between a Curve and a Line
4. The Negative Area Trap

If differentiation finds the gradient, integration finds the area. They are perfectly inverse operations — and CAIE tests both relentlessly. You absolutely cannot pass Additional Mathematics without mastering integration. This guide from our Ultimate Add Maths Guide gives you the exact mechanical process.

1. The Reverse Power Rule

Integration is the exact opposite of the Power Rule in differentiation. Instead of subtracting 1 from the power, you add 1. Instead of multiplying, you divide.

Rule: ∫ ax^n dx = ax^(n+1) / (n+1) + C

Example 1: ∫ 6x^2 dx = 6x^3/3 + C = 2x^3 + C

Example 2: ∫ 4x^3 + 2x dx = x^4 + x^2 + C

Example 3: ∫ 5 dx = 5x + C (a constant integrates to constant × x)

💡 Tutor's Tip

Never Forget +C! In indefinite integrals (no limits), you MUST write +C. It represents the unknown vertical shift. If C is missing, the examiner deducts 1 mark every single time, regardless of everything else being perfect.

2. Definite vs Indefinite Integrals

Indefinite (No Limits)

∫ 3x^2 dx = x^3 + C. The result is a general function with an unknown constant.

Definite (With Limits)

∫ from 1 to 3 of 3x^2 dx = [x^3] from 1 to 3 = (3)^3 - (1)^3 = 27 - 1 = 26. No +C needed because it cancels out when you subtract.

3. Area Between a Curve and a Line

When the exam asks for the area enclosed between a curve and a straight line, you must: 1) Find their intersection points (set equations equal). 2) Integrate (upper curve - lower curve) between those limits.

Key Formula

Area = ∫ from a to b of (f(x) - g(x)) dx, where f(x) is the upper curve and g(x) is the lower curve between the intersection points a and b.

4. The Negative Area Trap

This is the most devastating mistake students make. When a curve dips below the x-axis, the definite integral gives a negative number for that section. If you blindly integrate across the entire range, the negative area cancels out part of the positive area, giving you a completely wrong total.

The Fix

1. Find where the curve crosses the x-axis (set y = 0). 2. Split your integral at those roots. 3. Calculate each section separately. 4. Take the absolute value of any negative result. 5. Add all the positive areas together.

📋 From the Desk of Joshua Ramirez

Integration as Anti-Differentiation Check:If you are unsure whether your integral is correct, differentiate your answer. You should get back the original function you started with. If ∫ 4x^3 dx = x^4 + C, then d/dx of x^4 + C = 4x^3. They match. This self-checking trick catches careless errors instantly.

Frequently Asked Questions

What is Integration?▼

The exact mathematical reverse of differentiation. It reconstructs the original function from its derivative and calculates areas under curves.

What is the Reverse Power Rule?▼

Add 1 to the power, divide by the new power: integral of ax^n = ax^(n+1)/(n+1) + C.

What is a Definite Integral?▼

An integral with upper and lower limits that produces a specific numerical answer (the exact area) rather than a general function.

What happens when the curve goes below the x-axis?▼

The integral produces a negative value. You must split the integral at x-intercepts and add the absolute values of each section separately.

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The Area Machine: Mastering Definite Integration