Double Integral Calculator with steps: rectangular and polar coordinates

The double integral calculator that we present here is an excellent tool to solve all kinds of double integrals in rectangular or polar coordinates.

Double Integral Calculator


Use inf for +∞ and -inf for -∞    




Instructions for using the Double Integral Calculator

As you can see, the calculator has a very intuitive interface which makes it easy to use. To use it, you just have to follow the following steps:

  1. Choose the type of coordinates you will use to compute the double integral: Choose the “Rectangular” option to compute double integrals over rectangular regions, or select the “Polar” option to compute double integrals in polar coordinates.
  2. Select the differential of integration: if you have selected rectangular coordinates you have two drop-down lists to do so. In the case that you have selected polar coordinates, the integration differential will be rdrdt, where the variable t refers to the Greek letter theta.
  3. Enter into the calculator the function that will be the integrand of the double integral. To do this, keep in mind the table of valid functions presented in this section.
  4. Then enter the limits of integration, which can be numeric or mathematical expressions that use the variables present in the differential of integration. Note that to enter the number pi, you must write pi and if you want to write ∞ you must write inf.
  5. Select the decimal places of precision.
  6. Finally press the “Calculate” button to get the result. The solution will be deployed automatically showing the steps of the integration process.

For examples of double integrals you can press the “Examples” button.

Valid functions and symbols Description
sqrt() Square root
ln() Natural logarithm
log() Logarithm base 10
^ Exponents
abs() Absolute value
sin(), cos(), tan(), csc(), sec(), cot() Basic trigonometric functions
asin(), acos(), atan(), acsc(), asec(), acot() Inverse trigonometric functions
sinh(), cosh(), tanh(), csch(), sech(), coth() Hyperbolic functions
asinh(), acosh(), atanh(), acsch(), asech(), acoth() Inverse hyperbolic functions
pi PI number (π = 3.14159...)
e Neper number (e= 2.71828...)
i To indicate the imaginary component of a complex number.

What is a Double Integral?

Double integrals are all those integrals of functions in two variables over a rectangular region R2. A double integral represents the volume enclosed between a rectangular region R and a surface z=f(x,y) if f(x,y)>0. The double integral of a function of two variables, say f(x, y) over a rectangular region is represented by the following notation:

R  f(x,y) dA

f(x,y) dy dx

Double integrals are commonly used in physics, engineering, and other fields to solve problems involving the distribution of quantities over a region in two-dimensional space. For example, a double integral can be used to calculate the mass of an object by integrating its density over the volume of the object.

Properties of Double Integral

  • The double integral is linear:

The double integral is linear

  • The double integral is additive over rectangles that have at most one line segment in common:

The double integral is additive

  • If f(x,y)≤g(x,y) holds, we have:

Consequently, if f(x,y)≥0 at almost all points in R,

and if f(x,y)≤0 at almost all points of R,

  • For any function f integrable in R,

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