Integral Calculator & Area Under Curve (Fundamental Theorem & Solids of Revolution)
Compute definite integrals, geometric areas with automatic sign-change root splitting, enclosed areas between two functions, and volume of solids of revolution with interactive 2D graph and a 4-level trainer.
2D-Integrationsfläche & Kurvenverlauf
Summe aller Beträge (3 Teilflächen)
Netto-Flächenbilanz mit Vorzeichen
Vorzeichenwechsel-Trennpunkte
Vollständiger mathematischer Lösungsweg & Formeln
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Curved Areas & Barrel Volumes
In 1612, Johannes Kepler bought wine barrels for his wedding and was frustrated by merchants' inaccurate estimates for curved barrels. Archimedes had previously sought exact methods to determine areas bounded by parabolas and circles.
2. Fundamental Theorem of Calculus
Newton and Leibniz discovered the fundamental connection: integration (summing infinite thin slices) is the exact inverse of differentiation (finding slopes). Kepler pioneered slicing solids into infinitesimal discs.
3. Structural Engineering, Quantum Mechanics & Energy
From calculating battery electrical work to airplane wing lift distribution and quantum wave function probabilities, modern technology rests on integrals.
Real-World Applications of Integral Calculus
Integration is fundamental for accumulating quantities, physics kinematics, structural volumes, and financial economics.
Total distance from variable velocity
The accumulated distance s(t) of any moving body is the definite integral of its velocity function v(t) over time.
Integrating acceleration a(t) yields velocity v(t); a second integral computes total displacement s = ∫ v(t) dt.
Consumer & Producer Surplus
Integrals between market demand and supply curves measure total economic welfare at market equilibrium.
Consumer surplus equals the area between demand curve p(x) and equilibrium price p* on [0, x*].
Volume of Revolution Tanks
The storage capacity of cylindrical, conical, or parabolic storage vessels is computed via V = π ∫ [f(x)]² dx.
A cooling tower with contour f(x) has concrete volume calculated exactly through definite integration.
Mathematical Guide to Integral Calculus
Integral calculus reverses differentiation and enables exact calculations of curved areas, averages, and physical quantities.
Fundamental Theorem of Calculus (FTC)
If f is continuous on [a, b] and F is an antiderivative with F'(x) = f(x), then ∫_a^b f(x) dx = F(b) - F(a).
Geometric Area with Root Splitting
When roots x_i lie inside [a, b], the integral splits at each root to sum absolute values: A = ∫_a^(x_1) |f(x)| dx + ∫_(x_1)^b |f(x)| dx.
Area Between Two Curves
The area enclosed by two curves between intersection bounds is given by A = ∫_(x_1)^(x_2) |f(x) - g(x)| dx.
Solid of Revolution Volume (x-Axis)
Rotating f(x) about the x-axis on [a, b] produces a solid of volume V = π · ∫_a^b [f(x)]² dx.
Frequently Asked Questions (FAQ) on Integral Calculus
What is the difference between a definite integral and geometric area?▼
A definite integral yields net signed area (areas below the x-axis cancel areas above). Geometric area takes absolute magnitudes of all sub-areas.
How do you find the antiderivative of a polynomial term?▼
Apply the power rule of integration: x^n becomes (1 / (n + 1)) x^(n + 1). For example, 3x² becomes x³ + C.
Why do we need to find roots for area calculation?▼
Functions cross the x-axis at roots, changing sign. Splitting at roots prevents positive and negative areas from canceling each other out.
How do you find the enclosed area between two parabolas?▼
1. Set f(x) = g(x) to find intersection points. 2. Define d(x) = f(x) - g(x). 3. Integrate d(x) between intersections and take the absolute value.