Surface Integral & Related Calculus Calculator

Compute surface integrals, double integrals, line integrals, and flux integrals with step-by-step solutions.

Calculator Type

Function f(x,y,z)

Surface Type

x Lower Bound

x Upper Bound

y Lower Bound

y Upper Bound

z Function (if plane)

Radius (for sphere/cylinder)

u/v Parameter

v Parameter

Integration Order

Region Type

y Lower Bound (function)

y Upper Bound (function)

Curve Type

Parameter t range

Parametric Equations

r Lower Bound

r Upper Bound

θ Lower Bound

θ Upper Bound

In polar coordinates: dA = r dr dθ, x = r cosθ, y = r sinθ

r Lower Bound

r Upper Bound

θ Lower Bound

θ Upper Bound

z Lower Bound

z Upper Bound

In cylindrical coordinates: dV = r dr dθ dz, x = r cosθ, y = r sinθ

Result

Step-by-Step Solution

Select a calculator type and enter your function, then click "Calculate Integral" to see the step-by-step solution.

Surface Integral Calculator

Compute surface integrals over parametric surfaces, spheres, cylinders, and planes. Includes surface area and flux calculations.

Formula: ∬_S f(x,y,z) dS

Double Integral Calculator

Evaluate double integrals over rectangular and function-defined regions. Step-by-step integration with respect to x and y.

Formula: ∬_R f(x,y) dA

Surface Integral Vector Field

Calculate flux of vector fields through surfaces. Essential for electromagnetism and fluid dynamics.

Formula: ∬_S F · dS = ∬_S F · n dS

Line Integral Calculator

Compute line integrals along curves and paths. Work done by force fields and circulation calculations.

Formula: ∫_C F · dr

Polar Integral Calculator

Evaluate double integrals in polar coordinates. Perfect for circular and radial symmetry problems.

Formula: ∬ f(r,θ) r dr dθ

Flux Integral Calculator

Calculate flux through surfaces. Measures flow of vector fields across surfaces.

Formula: Φ = ∬_S F · n dS

Cylindrical Integral Calculator

Compute triple integrals in cylindrical coordinates. Ideal for problems with axial symmetry.

Formula: ∭ f(r,θ,z) r dr dθ dz

Surface Integral Visualization

Visualize surfaces and understand the geometric interpretation of surface integrals.

Concept: Summing function values over a surface weighted by area elements.

Common Integral Formulas

Surface Integral (z = f(x,y))

∬_S g(x,y,z) dS = ∬_D g(x,y,f(x,y)) √(f_x² + f_y² + 1) dA

Flux (Vector Field)

∬_S F · dS = ∬_D F · (-f_x, -f_y, 1) dA

Polar Coordinates

x = r cosθ, y = r sinθ, dA = r dr dθ

Cylindrical Coordinates

x = r cosθ, y = r sinθ, z = z, dV = r dr dθ dz

Line Integral

∫_C F · dr = ∫_a^b F(r(t)) · r'(t) dt

Sphere Surface Area

Surface Area = 4πR², dS = R² sinφ dφ dθ