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
Details about the region
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
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
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
See also Multivariable Calculus Notes - Chapter 15 - Sections 1-2 for more information.
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.
See Multivariable Calculus Notes - Chapter 15 - Sections 1-2 for more information.
(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:
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))
Polar Coordinates can be used with double integrals to simplify the evaluation process. Given a double integral:
We can make the following substitutions:
See Multivariable Calculus Notes - Chapter 15 - Sections 3-4 for more info.
See Multivariable Calculus Notes - Chapter 15 - Sections 3-4 for the following topics: