Double Integrals: Area or Volume?

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.
  • #1
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I'm reading in my Calculus book, and I see (I may see wrongly) that a double integral can describe both an Area and a Volume. Is that true? If that's true, how do I know when the Double Integral is describing an Area or a Volume? Thanks.
 
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  • #2
Well, it's sort of similar to how a single integral can be used to find both area and volume (and length, as well).
 
  • #3
The typical triple integral describing a volume of a certain domain [itex] \mathcal{D}\subseteq \mathbb{R}^{3} [/itex] is

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

Daniel.
 
  • #4
It depends on your problem. An integral doesn't necessarily describe something geometric, but it can be used to calculate surface areas, for example:

[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.
 
  • #6
[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?
 
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  • #7
Yeah,it was a typo by Galileo,but we usually write [itex] z=z\left(x,y\right) [/itex] when we indicate the equation of a surface in [itex] \mathbb{R}^{3} [/itex] explicitely .

Daniel.
 
  • #8
Yeah you are correct, that was one of my problems when I was first learning surface integrals... whenever I would get stuck, I would jump to the conclusion that I could simply use that general equation to solve the problem--- but it only works in specific cases. (Or when you're scrambling on a final!)
 

1. What is a double integral?

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.

2. How is a double integral different from a single integral?

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.

3. What are the different types of double integrals?

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.

4. What is the purpose of using a double integral?

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.

5. How do you solve a double integral?

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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