What is a Integral Calculator?
This calculator computes indefinite and definite integrals for finding area and accumulated quantities.
How it's calculated
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Definite integral: ∫[a,b] f(x)dx = F(b) − F(a)
Step-by-step example
Evaluating ∫[0,2] x² dx (the definite integral of x² from 0 to 2)
- Step 1 Find the antiderivative: ∫x² dx = x³/3 + C
- Step 2 Evaluate at the bounds: [x³/3]₀² = 2³/3 − 0³/3
- Step 3 Result: 8/3 ≈ 2.67
Interpretation: The area between the curve y=x² and the x-axis, from x=0 to x=2, is about 2.67.
Integration is 'stacking up infinitely thin pieces'
If differentiation is about breaking things apart, integration is about adding the broken pieces back together. To find the area under a curve, you can approximate it with thin rectangles — and as the width of each rectangle shrinks toward zero, the approximation error vanishes and you get the exact area. This is exactly why the area under a speed-vs-time graph gives total distance traveled, and the area under a power-vs-time graph gives total energy used.
Good to know
- Differentiation and integration are inverse operations — this is the Fundamental Theorem of Calculus, and the discovery that two seemingly unrelated problems (finding a tangent line, and finding an area) are actually two sides of the same coin is what made calculus so powerful.
- The ∫ symbol is an elongated 'S', standing for the Latin summa (sum) — the shape itself carries the meaning of 'adding things up.'
- Archimedes was already using integral-style ideas over 2,000 years ago — he calculated the area of a parabolic segment by filling it with infinitely many triangles (the method of exhaustion), centuries before calculus was formally developed.
- Forgetting the constant of integration (+C) is one of the most common mistakes — since differentiating any constant gives zero, you can never recover exactly what the original constant was, which is why C is always included.
- Not every function has an elementary antiderivative — e^(−x²) is a famous example that's easy to differentiate but has no closed-form integral, which is why normal distribution calculations rely on lookup tables or numerical methods instead.
Frequently asked questions
Q. Why is the constant of integration (C) needed?
Because differentiating any constant gives zero, the original constant can't be recovered — so C represents that unknown value.
Q. What's the difference between definite and indefinite integrals?
An indefinite integral is a general antiderivative (with +C); a definite integral evaluates to a specific number representing an area.
Q. Can the area come out negative?
Yes — regions below the x-axis are counted as negative. For actual physical area, take the absolute value.