Area under the parabola
- Input
- ∫ de 0 a 1 de x^2 dx
- Expected output
- 1/3 ≈ 0,333333
F = x³/3.
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definite integral calculator
In the Definite tab, enter the function and the limits a and b (pi, e, sqrt(2) and inf are accepted). The tool computes F(b) − F(a), compares it with an independent numerical integration and draws the region, with the positive and negative parts in different colors.
F = x³/3.
F = −cos x.
By power reduction: x/2 + sin 2x/4.
F = (x − 1)·eˣ.
Split at x = 0.
The indefinite integral of f is the family of functions F whose derivative is f. Since the derivative of a constant is zero, if F is an antiderivative then F + C is too, for any number C. That is why ∫ 2x dx = x² + C: x² + 1 and x² − 7 also have derivative 2x. The constant can only be pinned down with an extra condition, such as F(0) = 3.
The indefinite one is a family of functions, F + C. The definite one is a number, the signed area between the curve and the x axis between two limits. The second comes from the first through F(b) − F(a).
Yes: pi, e, sqrt(2), pi/2, 2*pi. Values such as sin(π/6) are handled with exact values, so the answer comes out symbolic, like π/4 instead of 0.785398.
Computers use 64-bit floating point, with about 16 decimal digits. Differences of the order of 1e-15 are rounding noise, not method error.
Differentiating the result
F′(x) = f(x) at 14 test points (maximum error 2.2e-16). Both expressions are also identical after expanding.
| x | f(x) | F′(x) | Difference |
|---|---|---|---|
| -2.26376 | -0.23533751 | -0.23533751 | 2.8e-17 |
| 0.565523 | 0.99552794 | 0.99552794 | 1.1e-16 |
| 4.44149 | 377.08727 | 377.08727 | 5.7e-14 |
| -0.222645 | -0.17820488 | -0.17820488 | 0 |
| -0.357476 | -0.25003303 | -0.25003303 | 0 |
| 0.460526 | 0.72989202 | 0.72989202 | 2.2e-16 |
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