Double Integral

A double integral is an integral of a function of multiple variables rather than a function of just a single variable. A double integral over the region is defined as the following:

Details about the region and other notation info can be found in multiple integral notation.

In a single variable integral the bounds are the range of integration along the specified axis, but in double integrals it now represents an area instead of just a range of numbers. This allows us to find the volume under the curve given by bounded by the area of region .

Using a double integral to find area

The above graph was generated using desmos using this graph.

#rewrite -- maybe? this needs to be proof read

As mentioned previously we can use double integrals to find the area between different curves and lines as long as we have their definitions or functions. As shown above, we've found that the area between and is equal to 4. When written as a double integral this looks like the following:

The upper and lower bounds of integration for the outermost integral are given by the points when our two lines connect, specifically the x coordinate (as is the outer of the two). The bounds of the inner integral are given by writing our two equations in terms of y. For example in out graph -x is the lower of our two equation thus it is the lower bound. Note that you cannot have y in your bounds as it breaks the integral.

Solving

To solve double integrals you break it into two different integrals, an inner and an outer one. You then can ignore the outer integral and solve the inner one like you would for a single variable integral.

Once we solve the inner integral we can then place it back in our outer integral and solve that one too.

Which gives us our final answer.

Solving Double Integrals using Python

(Using the SymPy Online Interpreter) You can find double integrals using SymPy's integrate function. The integrate function requires 3 bits of information: the function to integrate, the bounds of the inner integral, and the bounds of the outer integral. Given our previous example:

  • Our function to integrate is:
  • The bounds of the inner integral are:
  • The bounds of the outer integral are

Notice that we've specified the variable (x or y) that the bounds are related to, we will need to tell the integrate function which variable goes with which bounds.

Now that we have all the information we need we can call the integrate function. The function uses the following syntax: integrate(function to integrate, (inner var, lower bound, upper bound), (outer var, lower bound, upper bound)).

integrate(1, (y, -x, 2-x**2), (x, -1, 2))

Solving a double integral using python example

Double Integrals using Polar Coords

Polar Coordinates can be used with double integrals to simplify the evaluation process. Given a double integral:

We can make the following substitutions:

  • - Note the r added in the equality

Applications