Welcome to the **Step-by-Step Integral Calculator**. This tool helps you quickly find the definite integral of a common polynomial function ($ax^2 + bx + c$) between two specified bounds. Get accurate results and detailed calculation steps instantly.
Step-by-Step Integral Calculator
Calculate the definite integral of the function $f(x) = ax^2 + bx + c$ from $A$ to $B$.
Step-by-Step Integral Calculator Formula
The calculator uses the Fundamental Theorem of Calculus to solve the definite integral of a second-degree polynomial:
Where $F(x) = \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx$ is the antiderivative, and the final result is $F(B) – F(A)$.
Formula Source: Fundamental Theorem of Calculus (Wikipedia), Polynomial Integral (Wolfram MathWorld)
Variables
The inputs for this integral calculator are defined as:
- Coefficient of $x^2$ ($a$): The number multiplying the $x^2$ term.
- Coefficient of $x$ ($b$): The number multiplying the $x$ term.
- Constant Term ($c$): The non-variable term of the function.
- Lower Bound ($A$): The starting value for the integration interval.
- Upper Bound ($B$): The ending value for the integration interval.
Related Calculators
You may also find these related calculus and financial tools useful:
What is Step-by-Step Integral Calculation?
Integration is a fundamental concept in calculus used to find the total displacement, area, or volume. It is essentially the reverse process of differentiation (finding the derivative). Definite integration, specifically, finds the exact area under a curve between two points (the bounds), often representing accumulation over an interval.
The “step-by-step” process involves finding the antiderivative of the function first, then evaluating that antiderivative at the upper and lower bounds, and finally subtracting the lower bound value from the upper bound value. This is the core principle of the Fundamental Theorem of Calculus, which connects differentiation and integration.
How to Calculate a Definite Integral (Example)
Let’s calculate the integral of $f(x) = 2x^2 + 4x + 1$ from $A=0$ to $B=3$:
- Find the Antiderivative: Apply the power rule ($\int x^n dx = \frac{x^{n+1}}{n+1}$) to each term. $$\int (2x^2 + 4x + 1) dx = \frac{2}{3}x^3 + \frac{4}{2}x^2 + 1x = \frac{2}{3}x^3 + 2x^2 + x$$
- Evaluate at Upper Bound ($B=3$): Substitute $x=3$ into the antiderivative. $$F(3) = \frac{2}{3}(3)^3 + 2(3)^2 + 3 = \frac{2}{3}(27) + 2(9) + 3 = 18 + 18 + 3 = 39$$
- Evaluate at Lower Bound ($A=0$): Substitute $x=0$ into the antiderivative. $$F(0) = \frac{2}{3}(0)^3 + 2(0)^2 + 0 = 0$$
- Subtract and Find Result: Calculate $F(B) – F(A)$. $$\text{Result} = 39 – 0 = 39$$
Frequently Asked Questions (FAQ)
- What is the purpose of the bounds ($A$ and $B$) in integration?
- The bounds define the specific interval on the x-axis over which the area or accumulation is being measured. They are essential for definite integrals, but not for indefinite integrals (which result in a function plus a constant $C$).
- Can this calculator handle functions other than polynomials?
- This specific calculator module is designed for the simple polynomial form $ax^2 + bx + c$. For more complex functions (like trigonometric or exponential functions), specialized symbolic integration software is required.
- What does the ‘step-by-step’ feature show?
- The step-by-step feature details the application of the Fundamental Theorem of Calculus: finding the antiderivative, evaluating it at the upper bound, evaluating it at the lower bound, and calculating the final difference to arrive at the definite integral value.
- Why is integration important for engineering and finance?
- In engineering, integrals calculate centroids, moments of inertia, and total work done. In finance, they are used to calculate present value (accumulated interest), continuously compounded returns, and option pricing models.