Enter each component as an expression in t. Supported: +, -, *, /, ^ (power), and functions like sin(), cos(), exp(), log(), sqrt().
| Vector function F(t) | Bounds [a,b] | Integral ∫F(t)dt |
|---|---|---|
| <2t, sin(t), t²> | [0, π] | <π², 2, π³/3> ≈ <9.869604401, 2, 10.33542556> |
| <cos(t), exp(t), 1> | [0, 1] | <sin(1), e − 1, 1> ≈ <0.841470985, 1.718281828, 1> |
| <t³, √t> | [0, 4] | <16, 16/3> ≈ <16, 5.333333333> |
These examples use closed-form results; your calculator output should closely match with enough intervals.
For a vector-valued function F(t) = <f1(t), f2(t), …, fk(t)>, the definite integral is computed component-wise:
∫[a→b] F(t) dt = < ∫[a→b] f1(t) dt, ∫[a→b] f2(t) dt, …, ∫[a→b] fk(t) dt >.
n):h = (b − a)/n. Then
∫ f(t) dt ≈ (h/3)[ f(a) + f(b) + 4Σ f(a+(2i−1)h) + 2Σ f(a+2ih) ].
∫ f(t) dt ≈ h[ 0.5f(a) + Σ f(a+ih) + 0.5f(b) ].∫ f(t) dt ≈ h Σ f(a+(i+0.5)h).±1/√3 after mapping to the subinterval.- Select whether your vector has 2 or 3 components.
- Enter
f1(t),f2(t), and optionallyf3(t). - Set bounds
aandb, then pick a method and interval count. - Press Submit to compute the integral vector and related values.
- Use Download CSV or Download PDF to save your report.
1) What does it mean to integrate a vector function?
You integrate each component separately over the same interval. The result is another vector whose components are the definite integrals of the original components.
2) Does this compute indefinite integrals?
This tool focuses on definite integrals using numerical methods. For symbolic antiderivatives, use a computer algebra system and then evaluate the result at your bounds.
3) Which variable should I use in expressions?
Use t as the variable name. Write expressions like sin(t), 2*t, or t^2.
4) What functions and constants are supported?
Common trig and inverse trig, exp, log, sqrt, abs, plus pi and e. You can also use sec, csc, and cot.
5) How do I choose a good interval count?
Start with 200–1000 for smooth functions. Increase n when curves oscillate, have sharp bends, or when results change noticeably between runs.
6) Why does Simpson’s method require an even number of intervals?
Simpson’s rule fits parabolas over pairs of subintervals. That pairing requires an even interval count so the full range can be covered by these two-step segments.
7) What is “integral of magnitude” and why is it different?
∫||F(t)||dt adds the length of the vector along the interval, while ||∫F(t)dt|| takes the length after integrating. They match only in special cases.
8) How are CSV and PDF downloads produced?
Downloads are generated in your browser from the computed report. CSV stores summary and samples in rows, while PDF produces a printable page with the same information.