Definite Integral Calculator (Polynomial)
Exact definite integral of a cubic polynomial.
Enter your values
Enter the coefficient of x^3 you are working with.
Enter the coefficient of x^2 you are working with.
Enter the coefficient of x you are working with.
Enter the constant you are working with.
Enter the lower bound you are working with.
Enter the upper bound you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Definite integral
- 2.25
- Antiderivative F(x)
- 0.25x^4 + 0x^3 + -2x^2 + 0x
- F(upper) - F(lower)
- 2.25 - 0
Quick answer
Integrate term by term using x^(n+1)/(n+1), then evaluate at the upper bound minus the lower bound.
∫[a,b] f(x)dx = F(b)-F(a)
At a glance
| What it does | Exact definite integral of a cubic polynomial. |
|---|---|
| Category | Math |
| Inputs needed | Coefficient of x^3, Coefficient of x^2, Coefficient of x, Constant, Lower bound, Upper bound |
| Main output | Definite integral |
| Formula | ∫[a,b] f(x)dx = F(b)-F(a) |
| Cost | Free — no sign-up, no download |
How to use the Definite Integral Calculator (Polynomial)
- 1Enter the polynomial coefficients up to cubic (set unused ones to 0).
- 2Enter the lower and upper bounds of integration.
- 3The tool integrates term by term and subtracts F(lower) from F(upper).
Inputs explained
- Coefficient of x^3
- Enter the coefficient of x^3 you are working with.
- Coefficient of x^2
- Enter the coefficient of x^2 you are working with.
- Coefficient of x
- Enter the coefficient of x you are working with.
- Constant
- Enter the constant you are working with.
- Lower bound
- Enter the lower bound you are working with.
- Upper bound
- Enter the upper bound you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Coefficient of x^31
- Coefficient of x^20
- Coefficient of x-4
- Constant0
- Lower bound0
- Upper bound3
Results
- Definite integral2.25
- Antiderivative F(x)0.25x^4 + 0x^3 + -2x^2 + 0x
- F(upper) - F(lower)2.25 - 0
Frequently asked questions
What does a negative integral mean?
The curve lies below the x-axis on average over that interval.
Can I integrate non-polynomial functions?
Use the numerical Simpson's rule tool for functions without a simple closed form here.
What do I need to enter into the Definite Integral Calculator (Polynomial)?
Just 6 values: coefficient of x^3, coefficient of x^2, coefficient of x, constant, lower bound, upper bound. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Definite Integral Calculator (Polynomial) work out the answer?
Integrate term by term using x^(n+1)/(n+1), then evaluate at the upper bound minus the lower bound. It applies the formula ∫[a,b] f(x)dx = F(b)-F(a) and shows the working so you can check each step by hand.
Is the Definite Integral Calculator (Polynomial) free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.