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In summary, the conversation discusses the use of double integrals to describe both area and volume. It is mentioned that a single integral can also be used for this purpose, and that the specific function used in the integral depends on the problem at hand. Additionally, there is a brief mention of surface integrals and their applications.

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[tex] V_{D}=\iiint_{\mathcal{D}} \ dV [/tex]

Daniel.

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[tex]\iint_D \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial g}{\partial y}\right)^2}dA[/tex]

or

[tex]\iint_D 1 dA[/tex]

or volumes:

[tex]\iint_R f(x,y)dA[/tex]

It depends on the problem.

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ok thanks.

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[tex]\iint_D \sqrt{1+\left(\frac{\partial g}{\partial x}\right)^2+\left(\frac{\partial g}{\partial y}\right)^2}\partial A[/tex]

correct?

correct?

Last edited:

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Daniel.

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A double integral is a mathematical concept used to calculate the area or volume of a three-dimensional shape by integrating a function over a certain region in the shape.

A single integral calculates the area under a curve in a two-dimensional space, while a double integral calculates the volume under a surface in a three-dimensional space.

The two main types of double integrals are iterated integrals and double integrals with polar coordinates. Iterated integrals involve integrating one variable at a time, while double integrals with polar coordinates use polar coordinates to integrate over circular or symmetrical regions.

A double integral has many applications in mathematics and science, such as calculating the volume of a solid, finding the mass of an object with varying density, and determining the center of mass of an object.

To solve a double integral, you first need to determine the limits of integration for both variables. Then, you can evaluate the integral using various integration techniques, such as substitution, integration by parts, or partial fractions. It is also important to understand the type of region being integrated and choose the appropriate method for calculating the integral.

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