Use this calculator to find the definite integral of a simple quadratic function and see the detailed steps using the Fundamental Theorem of Calculus.
Integral Calculator Steps
Calculated Integral Value
0.00Integral Calculator Steps Formula
This calculator determines the value of a definite integral for a quadratic function, which follows the standard polynomial integration rules.
General Form of Definite Integral:
$$ \int_{L}^{U} (A x^2 + B x + C) \,dx $$Integration (Antiderivative) using the Power Rule:
$$ F(x) = \frac{A}{3} x^3 + \frac{B}{2} x^2 + C x $$Final Result using the Fundamental Theorem of Calculus:
$$ \text{Result} = F(U) – F(L) $$Variables
- Coefficient A: The multiplier for the $x^2$ term in the function.
- Coefficient B: The multiplier for the $x$ term in the function.
- Constant Term C: The non-variable term in the function.
- Lower Limit ($L$): The starting point of the integration range.
- Upper Limit ($U$): The ending point of the integration range.
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What is integral calculator steps?
Integration is a fundamental concept in calculus used to find the total sum, area, or volume of a continuous range. It is essentially the reverse process of differentiation (finding the derivative). When you calculate an integral with steps, you are following a rigorous mathematical procedure to find the antiderivative and then applying the limits of integration.
The definite integral, which this calculator solves, represents the signed area of the region bounded by the function’s graph, the x-axis, and the vertical lines defined by the upper and lower limits. Understanding the steps—applying the power rule, finding the antiderivative $F(x)$, and substituting the limits $F(U) – F(L)$—is crucial for mastering calculus and its applications in physics, engineering, and finance.
How to Calculate integral calculator steps (Example)
Let’s use the function $f(x) = 2x^2 + 4x + 1$ with a Lower Limit ($L$) of 1 and an Upper Limit ($U$) of 3.
- Identify the function and limits: $A=2, B=4, C=1, L=1, U=3$.
- Find the antiderivative $F(x)$: Apply the Power Rule $\int x^n dx = \frac{x^{n+1}}{n+1}$. $$ F(x) = \frac{2}{3} x^3 + \frac{4}{2} x^2 + 1x = \frac{2}{3} x^3 + 2 x^2 + x $$
- Evaluate $F(U)$: Substitute the Upper Limit ($U=3$) into $F(x)$: $$ F(3) = \frac{2}{3} (3)^3 + 2 (3)^2 + 3 = 18 + 18 + 3 = 39 $$
- Evaluate $F(L)$: Substitute the Lower Limit ($L=1$) into $F(x)$: $$ F(1) = \frac{2}{3} (1)^3 + 2 (1)^2 + 1 = 0.6667 + 2 + 1 = 3.6667 $$
- Calculate the final integral: Subtract $F(L)$ from $F(U)$. $$ \text{Result} = F(3) – F(1) = 39 – 3.6667 = 35.3333 $$
Frequently Asked Questions (FAQ)
How does the Fundamental Theorem of Calculus relate to these steps?
The Fundamental Theorem of Calculus states that if $F$ is an antiderivative of $f$, then the definite integral from $L$ to $U$ is $F(U) – F(L)$. Our steps follow this theorem precisely: we find the antiderivative and then evaluate it at the limits to get the final area under the curve.
Can this calculator handle functions other than quadratics?
This specific implementation is designed for the general quadratic form ($Ax^2 + Bx + C$). While the core rules apply to any polynomial, non-polynomial functions (like trigonometric or exponential) require different integration techniques (e.g., substitution, integration by parts) not covered by this tool.
What if my lower limit is greater than my upper limit?
Mathematically, if $L > U$, the definite integral is the negative of the integral calculated from $U$ to $L$. The calculator handles this by simply substituting the provided limits, resulting in a negative value, which is correct in the context of signed area.
Why is integration used in real-world problems?
Integration is used to calculate accumulated quantities. Examples include: finding the total distance traveled from a velocity function, calculating the volume of irregular shapes, determining the work done by a variable force, and calculating probabilities in statistical distributions.