# Fundamental theorem of calculus

If F' = f, then the integral of f from a to b equals F(b) - F(a). Also, the derivative of the integral from a to x of f(t) dt is f(x).

Maths (university) · Part 05: Integration · Card 5 of 10 · Last updated 2026-10-19

## Analogy

A bridge: it links the slope world (derivatives) to the area world (integrals).

## Example

**Integral of 2x from 1 to 3**

F(x) = x^2.
F(3) = 9 and F(1) = 1.
Integral = 9 - 1 = 8.
Check: trapezoid area (2+6)/2 x 2 = 8.

## In practice

Turns area problems into easy subtraction once you have an antiderivative.

## Practice

### Flashcards

- **What is the integral of x^2 from 0 to 3?** 9.
- **State the fundamental theorem of calculus, part two.** The integral of f from a to b equals F(b) - F(a), where F' = f.

### Interview questions

**Evaluate the integral of 3x^2 + 2x from 1 to 2 and explain each step.**

- Antiderivative is x^3 + x^2
- Evaluate at 2: 8 + 4 = 12
- Evaluate at 1: 1 + 1 = 2
- Subtract: 12 - 2 = 10
- Fundamental theorem of calculus justifies it

## Questions people ask

### Why is the fundamental theorem so important?

It shows that integration and differentiation undo each other. Without it you would need to add up infinitely many rectangles for every area. With it, finding an antiderivative and subtracting two values gives the exact answer in a few lines.

### Do I need + C in a definite integral?

No. The C cancels when you subtract. F(b) + C minus F(a) + C leaves F(b) - F(a). So you can leave it out, and write the antiderivative in square brackets with the limits before substituting the top and bottom values.

## Related concepts

- [Power rule for integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-2)
- [Definite integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-4)
- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)
- [Riemann sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-3)

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