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# Double Integral Calculator

_{lower}= x

_{upper}=

_{lower}= y

_{upper}=

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## Double Integral Lesson

#### Lesson Contents

### What is a Double Integral?

The double integral calculator above computes the definite integral of your function with the x and y limits you provided. The region that it integrates over is a rectangle on the x-y plane. This 2-dimensional rectangle on the x-y plane extends upwards to the surface produced by f(x, y). The image below shows an example of a double integral with z = f(x, y) as a function of x and y.

**The solution of a double integral is the volume between the surface and the x-y plane, as bound by the rectangle defined by the x and y limits.** For example, the image above has x limits of 2 and 5, and y limits of 2 and 3. This produces the rectangle that rests on the x-y plane, is 3 units long, and 1 unit wide. The dark shaded region of the surface is the projection of the rectangle up to the surface.

### Double Integrals More in Depth

As we can see in the image above, z = f(x, y) is a function of x and y and therefore the function produces a surface with z values that vary with x and the y. The region is a rectangle with side lengths determined by the size of the integrating region for x and y. **There are an infinite number of infinitesimal columns that extend from the surface down to the x-y plane.** These extremely small columns are notated as *dxdy* or *dydx* depending on the order of integration that we choose when setting the problem up. The double integral calculator on this page uses the order *dxdy* because it simplifies your inputs.

When calculating a double integral by hand, we can choose either *dxdy* or *dydx* because either will get the correct solution. We simply must ensure that the order of the integral limits matches the order of *dxdy* or *dydx*. This is because **we perform a double integral sequentially, from inside to outside.** Therefore, our *x* limits should be used during the *dx* portion of the integral, and the *y* limits should be used during the *dy* portion of the integral.

Double and triple integrals both calculate volume in 3-dimensional space. We discussed earlier that double integrals calculate the volume between a region of a surface and a plane. Conversely, triple integrals calculate the volume between two surfaces that produce a continuous shape together.

In summary, double integrals calculate projected volume whereas triple integrals calculate the volume of a shape. A triple integral adds a third integration dimension, a *dz*, and a set of *z* limits. Please note that double integrals and triple integrals may use any set of variables as long as the variables correspond to the coordinate system being used.

## How the Double Integral Calculator Works

The calculator on this page computes your double integral symbolically by using a computer algebra system. In symbolic integration, the computer uses algebra and integral rules to take the antiderivative of the function before applying the fundamental theorem of calculus. In essence, symbolic integration follows the same steps as a human with a paper and pencil would. It has the capability to attain near perfect solution accuracy. The calculator on this page is accurate to a minimum of the 5th decimal place!

The alternative to using symbolic integration to solve integrals is called numerical methods/integration. A numerical routine performs a relatively small, approximated version of the problem as many times as necessary to converge to an accurate solution. Generally, numerical routines can solve a greater range of problems but can take longer and potentially be less accurate.

Result :

Worksheet 1

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Cheat sheet

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