Multiple Integrals Calculator
Calculate double and triple integrals over regions in 2D and 3D space
Category: Mathematics
Multiple Integrals Calculator Inputs
Multiple Integrals Calculator Formula
Equation
∬_R f(x,y)dA = ∫∫ f(x,y)dxdy or ∭_V f(x,y,z)dV
Excel Formula
=∬_Rf(x,y)dA=∫∫f(x,y)dxdyor∭_Vf(x,y,z)dV
Variables
- Integral Type — Choose the Integral Type option used by the Multiple Integrals Calculator.
- Function f(x,y) or f(x,y,z) — Enter the Function f(x,y) or f(x,y,z) text used by the Multiple Integrals Calculator.
- Region Description — Enter the Region Description text used by the Multiple Integrals Calculator.
How the Multiple Integrals Calculator Works
Multiple integrals extend single-variable integration to functions of several variables, enabling computation of areas, volumes, masses, and other multidimensional quantities. A **double integral** $\iint_R f(x,y) \, dA$ integrates over a 2D region $R$, computing signed volume under the surface $z = f(x,y)$. A **triple integral** $\iiint_V f(x,y,z) \, dV$ integrates over a 3D region $V$. The fundamental theorem—**Fubini's theorem**—allows reducing multiple integrals to iterated single integrals: $\iint_R f(x,y) \, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx$ (assuming continuity and proper region). Change of variables (generalized substitution) uses the **Jacobian determinant** to transform coordinates: $\iint_R f(x,y) \, dx \, dy = \iint_S f(x(u,v), y(u,v)) |J| \, du \, dv$ where $J = \frac{\partial(x,y)}{\partial(u,v)}$. Polar, cylindrical, and spherical coordinates simplify integration over circular, cylindrical, and spherical regions. Multiple integrals appear throughout physics (center of mass, moment of inertia), probability (joint distributions), engineering (heat flow, fluid dynamics), and economics (multi-dimensional optimization).
The core relationship is ∬_R f(x,y)dA = ∫∫ f(x,y)dxdy or ∭_V f(x,y,z)dV. Typical inputs include Integral Type, Function f(x,y) or f(x,y,z), Region Description.
Enter your values in the multiple integrals calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Multiple Integrals Calculator Theory & Explanation
Double Integrals over Rectangular Regions
For rectangular region R = [a,b] × [c,d], Fubini's theorem states: \iint_R f(x,y) \, dA = ∫_a^b ∫_c^d f(x,y) \, dy \, dx = ∫_c^d ∫_a^b f(x,y) \, dx \, dy The order of integration doesn't matter for continuous f over rectangles. Compute as iterated integral: integrate inner integral first (treating other variable as constant), then outer. Example: \iint_R xy \, dA over R = [0,1] × [0,2]. Compute ∫_0^1 ∫_0^2 xy \, dy \, dx = ∫_0^1 x [(y^2)/(2)]_0^2 dx = ∫_0^1 2x \, dx = [x^2]_0^1 = 1. Geometrically, this is the signed volume under z=xy over the rectangle. Double integrals compute area (f=1), average value ((1)/(\textArea)(R) \iint f), and mass (\iint \rho(x,y) dA for density \rho).
\iint_R f(x,y) \, dA = ∫_a^b ∫_c^d f(x,y) \, dy \, dx
Double Integrals over General Regions
For non-rectangular regions, limits depend on variables. **Type I** (vertically simple): R = \(x,y) : a ≤ x ≤ b, g_1(x) ≤ y ≤ g_2(x)\. Integrate: \iint_R f(x,y) \, dA = ∫_a^b ∫_g_1(x)^g_2(x) f(x,y) \, dy \, dx **Type II** (horizontally simple): R = \(x,y) : c ≤ y ≤ d, h_1(y) ≤ x ≤ h_2(y)\. Integrate: \iint_R f(x,y) \, dA = ∫_c^d ∫_h_1(y)^h_2(y) f(x,y) \, dx \, dy Choose order based on region shape. Example: triangle with vertices (0,0), (1,0), (1,1) is Type I with 0 ≤ x ≤ 1, 0 ≤ y ≤ x. Computing \iint_R dA = ∫_0^1 ∫_0^x dy \, dx = ∫_0^1 x \, dx = (1)/(2) (area of triangle). Changing order may simplify computation, especially if inner integral is difficult in one order but easy in the other.
\iint_R f \, dA = ∫_a^b ∫_g_1(x)^g_2(x) f(x,y) \, dy \, dx \quad \text(Type I)
Polar Coordinates for Double Integrals
For regions with circular symmetry, polar coordinates (r, θ) where x = r\cosθ, y = r\sinθ simplify integration. The area element transforms: dA = dx \, dy = r \, dr \, dθ (the r factor is the Jacobian). Double integral becomes: \iint_R f(x,y) \, dx \, dy = \iint_S f(r\cosθ, r\sinθ) \, r \, dr \, dθ Example: disk x^2 + y^2 ≤ 1. Region in polar: 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π. Compute \iint_R (x^2 + y^2) \, dA = ∫_0^2π ∫_0^1 r^2 · r \, dr \, dθ = ∫_0^2π [(r^4)/(4)]_0^1 dθ = ∫_0^2π (1)/(4) dθ = (π)/(2). The extra r in r \, dr \, dθ is crucial—forgetting it is a common error. Polar coordinates excel for circles, annuli, sectors, and spiral regions.
\iint_R f(x,y) \, dx \, dy = ∫_α^β ∫_r_1(θ)^r_2(θ) f(r\cosθ, r\sinθ) \, r \, dr \, dθ
Triple Integrals and Volume Computation
Triple integrals \iiint_V f(x,y,z) \, dV extend to 3D. For rectangular box V = [a,b] × [c,d] × [e,f]: \iiint_V f(x,y,z) \, dV = ∫_a^b ∫_c^d ∫_e^f f(x,y,z) \, dz \, dy \, dx Six possible orders of integration—choose based on difficulty. For f=1, the triple integral gives volume of V. Example: solid bounded by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1-x, 0 ≤ z ≤ 1-x-y (tetrahedron). Volume: ∫_0^1 ∫_0^1-x ∫_0^1-x-y dz \, dy \, dx = ∫_0^1 ∫_0^1-x (1-x-y) \, dy \, dx = ∫_0^1 ((1-x)^2)/(2) dx = (1)/(6). Triple integrals compute mass (\iiint \rho dV), moments, centers of mass, and moments of inertia in 3D. They're essential for physics and engineering applications involving 3D regions.
\iiint_V f(x,y,z) \, dV = ∫ ∫ ∫ f(x,y,z) \, dz \, dy \, dx
Cylindrical Coordinates
For solids with cylindrical symmetry (axis along z), use (r, θ, z) where x = r\cosθ, y = r\sinθ, z = z. Volume element: dV = r \, dr \, dθ \, dz. Triple integral: \iiint_V f(x,y,z) \, dx \, dy \, dz = \iiint_W f(r\cosθ, r\sinθ, z) \, r \, dr \, dθ \, dz Example: cylinder x^2 + y^2 ≤ 1, 0 ≤ z ≤ h. Volume: ∫_0^2π ∫_0^1 ∫_0^h r \, dz \, dr \, dθ = 2π · (1)/(2) · h = π h (familiar formula V = π r^2 h with r=1). Cylindrical coordinates simplify integrals over cylinders, cones, and paraboloids of revolution. The extra r factor (Jacobian) must not be forgotten.
\iiint_V f \, dV = ∫ ∫ ∫ f(r\cosθ, r\sinθ, z) \, r \, dr \, dθ \, dz
Spherical Coordinates
For spherical symmetry (center at origin), use (\rho, θ, \phi) where x = \rho\sin\phi\cosθ, y = \rho\sin\phi\sinθ, z = \rho\cos\phi. Here \rho is distance from origin, θ is azimuthal angle (same as polar/cylindrical), \phi is polar angle from positive z-axis. Volume element: dV = \rho^2 \sin\phi \, d\rho \, dθ \, d\phi. Triple integral: \iiint_V f(x,y,z) \, dx \, dy \, dz = \iiint_W f(...) \, \rho^2 \sin\phi \, d\rho \, dθ \, d\phi Example: sphere x^2 + y^2 + z^2 ≤ a^2. Volume: ∫_0^2π ∫_0^π ∫_0^a \rho^2 \sin\phi \, d\rho \, d\phi \, dθ = 2π · 2 · (a^3)/(3) = (4π a^3)/(3) (familiar formula). The factor \rho^2\sin\phi is the Jacobian. Spherical coordinates are ideal for spheres, cones, and problems with radial symmetry.
dV = \rho^2 \sin\phi \, d\rho \, dθ \, d\phi
Change of Variables and Jacobian
General coordinate transformation (u,v) \to (x(u,v), y(u,v)) changes integration via Jacobian: \iint_R f(x,y) \, dx \, dy = \iint_S f(x(u,v), y(u,v)) |J(u,v)| \, du \, dv where Jacobian determinant J = (\partial(x,y))/(\partial(u,v)) = \beginvmatrix (\partial x)/(\partial u) & (\partial x)/(\partial v) \\ (\partial y)/(\partial u) & (\partial y)/(\partial v) \endvmatrix. For polar: J = \beginvmatrix \cosθ & -r\sinθ \\ \sinθ & r\cosθ \endvmatrix = r. For 3D, (u,v,w) \to (x,y,z): Jacobian is 3 × 3 determinant. Change of variables simplifies integrals over complicated regions by transforming to simpler shapes (e.g., ellipse to circle). It's the multivariable generalization of substitution u = g(x), dx = du/g'(x) from single-variable calculus. Jacobian measures how volume scales under transformation.
J = (\partial(x,y))/(\partial(u,v)) = \det\beginpmatrix x_u & x_v \\ y_u & y_v \endpmatrix
Multiple Integrals Calculator Worked Examples
Worked Example
Inputs
- integralType: double-rect
- function: x*y
- region: 0≤x≤1, 0≤y≤2
Result: ∬_R xy dA = 1
Explanation
∫₀¹ ∫₀² xy dy dx = ∫₀¹ x[y²/2]₀² dx = ∫₀¹ 2x dx = 1
Area of Triangle
Inputs
- integralType: double-rect
- function: 1
- region: 0≤x≤1, 0≤y≤x
Result: Area = 1/2
Explanation
∫₀¹ ∫₀ˣ 1 dy dx = ∫₀¹ x dx = 1/2
Common Multiple Integrals Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Multiple Integrals homework and study
- Multiple Integrals design and analysis
Multiple Integrals Calculator FAQs
What is a double integral and what does it represent?
A double integral ∬_R f(x,y) dA integrates a function of two variables over a 2D region R. Geometrically, if f(x,y) ≥ 0, it represents the volume under the surface z=f(x,y) above region R. More generally, it computes the signed volume (positive where f>0, negative where f<0). Applications: area (set f=1), mass (f=density), average value (divide by area), center of mass, moments of inertia. For example, ∬_R 1 dA = Area(R). Double integrals extend single integrals from 1D to 2D, crucial for physics, engineering, and probability (joint distributions).
What is Fubini's theorem?
Fubini's theorem lets you compute multiple integrals as iterated single integrals. For rectangular region [a,b]×[c,d] and continuous f: ∬_R f(x,y) dA = ∫_a^b ∫_c^d f(x,y) dy dx = ∫_c^d ∫_a^b f(x,y) dx dy. Order doesn't matter! This reduces 2D integration to two 1D integrations. For non-rectangular regions, limits depend on variables (Type I/II regions), but principle holds: integrate inner variable first (with limits possibly depending on outer variable), then outer variable. Fubini extends to triple integrals and higher dimensions. It's the foundation for actually computing multiple integrals.
When should I use polar coordinates for double integrals?
Use polar coordinates when the region or integrand has circular symmetry. Indicators: (1) Region is a disk, annulus, sector, or defined by r=f(θ). (2) Integrand contains x²+y² (becomes r²). (3) Limits in rectangular are complicated but simple in polar. Transform: x=r cos θ, y=r sin θ, dA=r dr dθ. The extra "r" factor is crucial! Example: disk x²+y²≤1 in polar is 0≤r≤1, 0≤θ≤2π (simple limits). Integrating f=x²+y²=r² becomes ∫∫ r²·r dr dθ. Polar drastically simplifies circular region integrals—master this technique!
What is the Jacobian and why do we need it?
The Jacobian J=∂(x,y)/∂(u,v) is the determinant of the transformation matrix when changing coordinates. It measures how area/volume scales: a tiny rectangle du×dv in (u,v)-space becomes a region of area |J| du dv in (x,y)-space. When changing variables, ∬_R f(x,y) dx dy = ∬_S f(x(u,v),y(u,v)) |J| du dv. Examples: polar J=r (explains the extra "r" factor), cylindrical J=r, spherical J=ρ²sin φ. Without Jacobian, integration in new coordinates gives wrong answer. It's the multivariable analog of dx=du/g'(x) in substitution. Think: Jacobian compensates for coordinate system distortion.
How do I choose the order of integration?
For rectangular regions, order doesn't affect result (Fubini), so choose easier computation. For general regions: (1) Sketch the region. (2) Check if Type I (vertical lines: x constant, y varies between functions of x) or Type II (horizontal lines: y constant, x varies). (3) If both work, choose order making inner integral easier. Example: if inner integral involves difficult function but becomes simple after integration, that's likely better order. Sometimes one order has constant limits (easier!) while other has variable limits. Switching order can transform impossible integral into easy one—powerful technique called "reversing order of integration."
What's the difference between cylindrical and spherical coordinates?
Cylindrical (r,θ,z): extends polar to 3D by adding z. x=r cos θ, y=r sin θ, z=z. Good for cylinders, cones (z=kr), axis symmetry. dV=r dr dθ dz. Spherical (ρ,θ,φ): full radial symmetry from origin. x=ρ sin φ cos θ, y=ρ sin φ sin θ, z=ρ cos φ. Good for spheres, cones (φ=constant), radial dependence. dV=ρ² sin φ dρ dθ dφ. φ is angle from z-axis (0 to π). Key: cylindrical for axis symmetry, spherical for point symmetry. Both have "extra factors" (r for cylindrical, ρ² sin φ for spherical) from Jacobian—must include!
Can I forget the extra factors like "r" in polar or "ρ² sin φ" in spherical?
NO! Forgetting these is the #1 error in multiple integrals. They're not optional—they come from the Jacobian (coordinate transformation). Without them, you get wildly wrong answers. Example: volume of sphere radius 1 in spherical. WRONG: ∫∫∫ dρ dθ dφ = π·2π = 2π². RIGHT: ∫∫∫ ρ²sin φ dρ dθ dφ = 4π/3 (correct formula). The "r" in polar, "r" in cylindrical, and "ρ²sin φ" in spherical are part of the volume/area element—non-negotiable. Always write dV or dA explicitly in new coordinates including these factors. This ensures you don't forget them.
How are multiple integrals used in physics?
Multiple integrals compute 3D quantities: (1) Mass: M=∭_V ρ(x,y,z) dV for density ρ. (2) Center of mass: x̄=(1/M)∭ x·ρ dV, similarly ȳ, z̄. (3) Moments of inertia: I_z=∬ (x²+y²)ρ dA (rotation about z). (4) Electric/gravitational fields: integrate force contributions over charge/mass distributions. (5) Fluid flow: flux through surfaces. (6) Heat/wave equations: integrate sources over volumes. (7) Probability: ∬ f(x,y) dx dy = P(region) for joint PDF. Example: moment of inertia of uniform disk about center uses ∬ r²·ρ·r dr dθ (polar). Physics relies heavily on multiple integrals for any spatially distributed quantity.