5.25
5.25
0
3
1.75
1.5
0.625
5.25
5.25
0
3
1.75
1.5
0.625
The Definite Integral Calculator evaluates the exact definite integral of any polynomial up to degree three over a specified interval. Given a function $$f(x) = ax^3 + bx^2 + cx + d$$ and bounds $$[a, b]$$, the calculator applies the power rule to find the antiderivative and uses the Fundamental Theorem of Calculus to compute $$F(b) - F(a)$$.
The definite integral is one of the central concepts in calculus, representing the net signed area between a curve and the x-axis over a closed interval. The notation $$\int_a^b f(x)\,dx$$ was introduced by Leibniz, where the elongated S symbolizes summation and $$dx$$ indicates the variable of integration. This concept evolved from the method of exhaustion used by Archimedes to find areas of parabolic segments over two millennia ago.
For polynomial functions, integration is particularly elegant because the power rule provides exact closed-form antiderivatives. There is no need for numerical approximation methods such as the trapezoidal rule or Simpson's rule — the calculator computes the mathematically exact result. The power rule states that $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for any $$n \neq -1$$, and this rule applies term by term to polynomials.
Beyond pure area calculation, this tool also computes the average value of the function over the interval, defined as $$\frac{1}{b-a}\int_a^b f(x)\,dx$$. The Mean Value Theorem for Integrals guarantees that for any continuous function, there exists at least one point $$c$$ in $$[a, b]$$ where $$f(c)$$ equals this average value.
Definite integrals of polynomials appear throughout science and engineering. In kinematics, integrating a cubic velocity function yields displacement. In beam theory, the bending moment is related to load distributions through integration. In thermodynamics, work done by an expanding gas can be modeled as the integral of a polynomial pressure function. Enter your polynomial coefficients and integration bounds to obtain exact results instantly.
The calculator applies the power rule of integration to each term of the polynomial $$f(x) = ax^3 + bx^2 + cx + d$$.
Step 1: Find the antiderivative.
$$F(x) = \frac{a}{4}x^4 + \frac{b}{3}x^3 + \frac{c}{2}x^2 + dx$$
Step 2: Evaluate at bounds using the Fundamental Theorem of Calculus.
$$\int_{a_{\text{bound}}}^{b_{\text{bound}}} f(x)\,dx = F(b_{\text{bound}}) - F(a_{\text{bound}})$$
Step 3: Compute the average value.
$$\bar{f} = \frac{1}{b_{\text{bound}} - a_{\text{bound}}} \int_{a_{\text{bound}}}^{b_{\text{bound}}} f(x)\,dx$$
Each step uses exact arithmetic on the polynomial coefficients. No numerical approximation is involved.
The Definite Integral Value represents the net signed area between the polynomial curve and the x-axis over $$[a, b]$$. If the polynomial is entirely above the x-axis on that interval, the integral equals the geometric area. If parts of the curve dip below the axis, those portions contribute negative area.
F(b) and F(a) are the antiderivative evaluated at the upper and lower bounds respectively. Their difference is the definite integral.
The Average Value tells you the constant function height that would produce the same total area over the same interval. This is useful for understanding the "typical" magnitude of the function.
Inputs
Results
F(x) = x⁴/4 − x³ + x² + x. F(3) = 81/4 − 27 + 9 + 3 = 20.25 − 27 + 9 + 3 = 5.25. F(0) = 0. Integral = 5.25. Average = 5.25/3 = 1.75.
Inputs
Results
F(x) = (2/3)x³ + 3x. F(2) = 16/3 + 6 = 5.333 + 6 = 11.333. F(−1) = −2/3 − 3 = −3.667. Integral = 11.333 − (−3.667) = 15. Average = 15/3 = 5.
A definite integral $$\int_a^b f(x)\,dx$$ computes the net signed area between the graph of $$f(x)$$ and the x-axis from $$x = a$$ to $$x = b$$. It is evaluated by finding an antiderivative $$F(x)$$ and computing $$F(b) - F(a)$$.
Regions where the function is positive contribute positive area, while regions where the function is negative contribute negative area. The integral returns the algebraic sum of these contributions, which may be zero even if the function is nonzero, if positive and negative regions exactly cancel.
By convention, $$\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$$. The calculator computes $$F(b_{\text{bound}}) - F(a_{\text{bound}})$$, so swapping the bounds reverses the sign of the result, which is mathematically correct.
The power rule states $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. Applied term by term to a polynomial, each coefficient is divided by one more than its power. For example, $$\int 3x^2\,dx = 3 \cdot \frac{x^3}{3} = x^3 + C$$.
The average value of $$f$$ on $$[a, b]$$ is $$\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx$$. It represents the height of a rectangle with the same base width $$b - a$$ that has the same area as the region under the curve.
This calculator handles polynomials up to degree 3 (cubic). For higher-degree polynomials, the same power rule applies — you would add more coefficient inputs. The principle remains identical: integrate each term using $$\int x^n\,dx = x^{n+1}/(n+1)$$.
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