The integral was invented to solve one problem: how much area sits between a curve and the x-axis? Everything else — antiderivatives, +C, the Fundamental Theorem — grew from that geometric question.
Here's how to think about area under curves, from rectangles to exact answers.
The Intuition: Rectangles First
[Diagram: Area under curve] -- Definite integral = signed area between curve and x-axis. Above axis = positive. Below = negative. Riemann sums approximate with thin rectangles.
Imagine slicing the area under f(x) = x² from x = 0 to x = 3 into thin rectangles.
Each rectangle has:
- Height = f(x) at that point
- Width = Δx (small slice)
Total area ≈ sum of all rectangles
More rectangles → thinner slices → better approximation → exact answer as Δx → 0.
This is a Riemann sum, and the integral is its limit.
Exact Area: The Definite Integral
Area = ∫[a,b] f(x) dx = F(b) − F(a)
Where F is the antiderivative of f.
Example: Area under f(x) = x² from 0 to 3
- F(x) = x³/3
- F(3) = 27/3 = 9
- F(0) = 0
- Area = 9 − 0 = 9 square units
Example: Area under f(x) = 2x + 1 from 1 to 4
- F(x) = x² + x
- F(4) = 16 + 4 = 20
- F(1) = 1 + 1 = 2
- Area = 20 − 2 = 18 square units
(Sanity check: it's a trapezoid with heights 3 and 9, width 3 → (3+9)/2 × 3 = 18 ✓)
When the Curve Goes Below the X-Axis
Signed area counts below-axis regions as negative.
Example: ∫[−1,1] x dx
- F(x) = x²/2
- F(1) = 0.5, F(−1) = 0.5
- ∫ = 0.5 − 0.5 = 0
The positive triangle (0 to 1) cancels the negative triangle (−1 to 0).
For total area (always positive), split at x-axis crossings:
- ∫[0,1] x dx = 0.5 (above)
- ∫[−1,0] |x| dx = 0.5 (below, take absolute value)
- Total area = 1
Approximation Methods
When you can't find an antiderivative, approximate:
Riemann Sums (Left/Right/Midpoint)
Divide [a,b] into n rectangles. Use left endpoints, right endpoints, or midpoints for heights.
∫[0,2] x² dx with 4 rectangles (left):
| Rectangle | x | Height (x²) | Width | Area | |-----------|---|-------------|-------|------| | 1 | 0 | 0 | 0.5 | 0 | | 2 | 0.5 | 0.25 | 0.5 | 0.125 | | 3 | 1.0 | 1.0 | 0.5 | 0.5 | | 4 | 1.5 | 2.25 | 0.5 | 1.125 |
Left sum ≈ 1.75 (actual = 2.67 — underestimate)
Trapezoidal Rule
Use trapezoids instead of rectangles for better accuracy:
Area ≈ (Δx/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]
Simpson's Rule (Most Accurate)
Uses parabolas to approximate:
Area ≈ (Δx/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
| Method | 4 rectangles | Error vs exact | |--------|-------------|---------------| | Left Riemann | 1.75 | −34% | | Trapezoidal | 2.75 | +3% | | Simpson's | 2.67 | <0.1% |
The Trench Truth: On exams, if you're asked for an approximation, use the midpoint rule — it's more accurate than left or right sums and doesn't require memorizing the trapezoidal formula. If you're asked for exact area, use the definite integral.
Calculate exact areas with our integral calculator — enter the function and bounds for instant step-by-step solutions.
Further Reading
Related: Derivative Calculator | Quadratic Formula Calculator | Square Root Calculator | Statistics Calculator | Standard Deviation Calculator | Circle Calculator | Area Converter
Discussion
Loading comments...