Integral Calculus Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in integral calculus.

Antiderivatives and Indefinite Integrals - Theory & Concepts - Antiderivative Family

Understanding antiderivatives, indefinite integral notation, basic integration formulas, and properties of linearity.

Simulation: Family of Antiderivatives

Constant (C)0.0

Legend

f(x): Original Function

F(x) + C: Antiderivative

Other Family Members

Observe how adjusting the constant C shifts the curve vertically without altering its shape or instantaneous slope.

Antiderivatives and Indefinite Integrals - Theory & Concepts - U Substitution

Understanding antiderivatives, indefinite integral notation, basic integration formulas, and properties of linearity.

U-Substitution: Coordinate Mapping & Area Equivalence

Visualizing the reverse Chain Rule. Watch how the coordinate stretching factor $du = g'(x)dx$ compresses/stretches the area elements between the original $x$ space and the substituted $u$ space.

\int_0^2 3x^2 \cos(x^3) \, dx \quad \Longleftrightarrow \quad \int_0^8 \cos(u) \, du

Sweep Coordinate (

x

): 1.20

x = 0x = 2

Coordinate Mapping: $u = x^3$

Mapped point $u$:1.7280

Stretch factor $du/dx = 3x^2$:4.320

∫ 3x² cos(x³) dx Area:0.98767

∫ cos(u) du Area:0.98767

The areas under both curves are exactly equal at every point! The $x$-space integrand is compressed on the left but stretched vertically by $3x^2$, matching the right $u$-space integral perfectly.

Original X-Space: $\int 3x^2\cos(x^3)dx$

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x-axis range [0, 2]

Transformed U-Space: $\int \cos(u)du$

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u-axis range [0, 8] ($u = x^3$)

Definite Integrals - Theory & Concepts - Riemann Sum

Understanding Riemann sums, the Fundamental Theorem of Calculus, properties of definite integrals, and improper integrals.

Simulation: Riemann Sums & Area

Rectangles (n)4

Coarse (2)Fine (100)

Area Analysis

Approximation:1.7500

Exact Integral:2.6666

Absolute Error:0.9166

As $n \to \infty$, the width $\Delta x \to 0$, and the Riemann sum converges precisely to the definite integral.

Definite Integrals - Theory & Concepts - Improper Integral

Understanding Riemann sums, the Fundamental Theorem of Calculus, properties of definite integrals, and improper integrals.

Improper Integral Visualization: ∫ (1/x²) dx from 1 to t

Upper bound t: 2.0

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Area Calculation:

Area(t) = 1 - 1/t

Current Area = 0.5000

As t → ∞, the area approaches 1. The integral converges to 1.

Techniques of Integration - Theory & Concepts - Integration By Parts

Mastering methods for evaluating complex integrals: u-substitution, integration by parts, trigonometric integrals, and partial fractions.

Interactive Lab: Integration by Parts (LIATE)

Current Problem:

\int x e^x \, dx

LIATE Priority List

LLogarithmic

IInverse Trig

AAlgebraic

TTrigonometric

EExponential

The function type highest on this list should be selected as u.

Which part should be u?Select the function based on the LIATE rule.

Techniques of Integration - Theory & Concepts - Trig Substitution

Mastering methods for evaluating complex integrals: u-substitution, integration by parts, trigonometric integrals, and partial fractions.

Trigonometric Substitutions

Visualizing the right triangle geometry

Reference Triangle

a

x

\sqrt{a^2 - x^2}

Target Radical:

\sqrt{a^2 - x^2}

\text{Let } x = a \sin \theta

Radical Simplification

Click "Next Step" to simplify the radical algebraically.

Techniques of Integration - Theory & Concepts - Wallis Formula

Mastering methods for evaluating complex integrals: u-substitution, integration by parts, trigonometric integrals, and partial fractions.

Wallis' Formula Calculator

Power (n): 4

Integral to Evaluate

\int_0^{\pi/2} \sin^{4} x \, dx

Applying Wallis' Formula (n is Even)

= \frac{3 \cdot 1}{4 \cdot 2} \cdot \frac{\pi}{2}

= \frac{3}{8} \cdot \frac{\pi}{2}

Decimal Result

\approx

0.5890

Applications of Integration - Theory & Concepts - Area Between Curves

Exploring the applications of definite integrals: finding the area between curves, calculating the volume of solids of revolution, determining arc length, finding the center of mass, fluid pressure, work, and surface area.

Area Between Curves Visualizer

Upper Curve f(x):

Lower Curve g(x):

The shaded region represents the area where f(x) ≥ g(x).

Applications of Integration - Theory & Concepts - Solid Revolution

3D Viewer: Solids of Revolution

Rotate 2D areas around the X-axis to generate volumes.

Disk Method: Area under Curve

Curve: $y = \sqrt{x}$

V = π ∫ (R(x))² dx

Drag to rotate, scroll to zoom

Applications of Integration - Theory & Concepts - Volume Slicing

Solid of Revolution: Visualization of Slices

Number of slices: 5

Disk Method Integral Concept:

V = π ∫ [R(x)]² dx

As the number of slices approaches infinity, the discrete disks/washers perfectly approximate the volume of the solid.

Applications of Integration - Theory & Concepts - Arc Length

Arc Length Approximation

Adjust the number of line segments to see how the Riemann sum approximates the exact curve length.

Number of Segments ($n$): 2

Approximation

412.31

Exact Length (Limit)

431.62

Applications of Integration - Theory & Concepts - Pappus Theorem

Theorems of Pappus: Torus Generator

Circle Radius (r): 2.0

Distance to Axis (

\bar{r}

): 5.0

Area of Circle (A):12.57

Circumference (L):12.57

Volume (

V = 2\pi\bar{r}A

):394.78

Surface Area (

S = 2\pi\bar{r}L

):394.78

Multiple Integrals - Theory & Concepts - Double Integral

Extending the concept of integration to functions of two or three variables: evaluating double and triple integrals, changing coordinate systems, and exploring their applications.

Double Integral & Volume Visualizer

Function $z = f(x, y)$

Grid Resolution (n x n): 10 x 10

Increase resolution to see the Riemann sum converge to the exact volume. Note: High resolution may affect performance.

Calculated Volume (Approx.)

25.190

V \approx \sum_{i=1}^n \sum_{j=1}^n f(x_i^*, y_j^*) \Delta A

Drag to rotate • Scroll to zoom

Multiple Integrals - Theory & Concepts - Polar Double Integral

Extending the concept of integration to functions of two or three variables: evaluating double and triple integrals, changing coordinate systems, and exploring their applications.

Polar Coordinates: Double Integrals Visualizer

Understand why the polar area element is $dA = r \, dr \, d\theta$ . Sweeping further from the origin enlarges the wedge area, proving that the extra $r$ factor represents coordinate scaling.

Select Integrand Function $f(r, \theta)$

Radius Limit (

R_{max}

): 3 m

Angle Limit (

\theta_{max}

): 180°

Wedge Element

dA

radius

r

: 2.0 m

Double Integral Formulation

\iint_R f(r, \theta) \, dA = \int_0^{\theta_{max}} \int_0^{R_{max}} r \, dr \, d\theta

Wedge Area dA:0.1200 m²

Integral Value:14.1372 m²

Polar Sweeping Grid

The Polar Area Element

In Cartesian coordinates, the differential area is a constant rectangle: $dA = dx \\, dy$ .

In Polar coordinates, the area element is a curved wedge of radial thickness $dr$ and arc length $r \\, d\\theta$ .

Therefore, the area of the wedge is:

dA = (r \\, d\\theta) \\cdot dr = r \\, dr \\, d\\theta

Drag the slider and watch how $dA$ (the red wedge) grows physically larger as you slide $r$ outwards. Closer to the origin, wedges are tightly squeezed; further out, they expand. The extra $r$ factor mathematically offsets this geometrical widening!