Exploring Multivariable Calculus: Double Integrals & Beyond

In summary, the use of double integrals in multivariable calculus allows for more complex problems to be solved, such as finding mass from density. While single integrals can be used for simpler problems, the double integral is necessary for more complicated regions. Additional useful applications of integrals in multivariable calculus include finding volumes, areas of revolution, and concentric shells. For more information on this topic, a helpful resource is available at http://www.math.hmc.edu/calculus/tutorials/.
  • #1
PrudensOptimus
641
0
OK OK I know Double Integral is from Multivar Calculus,

I was just wondering what we use it for... I heard is good for volumes, but can't yhou also find volumes by just 1 integral?

And also, aside from integrals in Multivar calc, what else are useful?

I want to get a intro to it, can anyone give a lecture or link to a intro page? Thanks.
 
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  • #2
While you're correct that a volume, or area, can be obtained from a single integral, the more general double integral is better suited to more complex problems.

It allows you to integrate a function over an area, while the triple integral does the same over a volume. There are really an infinite number of applications, but a good one is finding mass from density.

Using the double integral one can take the density function and integrate it over an area, thus finding the mass of a lamina. With a triple integral, you can find the mass of any relatively simple solid region, for which a density function exists.

If you'd like to see more about this, I've had good luck here:
http://www.math.hmc.edu/calculus/tutorials/
 
  • #3
No, you cannot calculate volumes with just one integral, if you are speaking of the volume of an arbitrary-shaped region. (Highly symmetric regions can be reduced to single integrals, because you can implicitly pull out some of the integrals as constants.)
 
  • #4
I was referring to the method of finding volume-using areas of revolution or concentric shells. Which of course does only apply to highly symmetric shapes. I assume this is what you're referring to.
 

1. What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with the study of functions of two or more variables. It extends the concepts of single-variable calculus to functions with multiple variables, allowing for the analysis of more complex systems and phenomena.

2. What are double integrals?

Double integrals are a type of integral in multivariable calculus that allows for the calculation of the volume under a surface in three-dimensional space. It involves integrating a function over a two-dimensional region, and the result is a number that represents the volume of the region under the surface.

3. How are double integrals different from single integrals?

While single integrals involve integrating a function over a one-dimensional interval, double integrals involve integrating a function over a two-dimensional region. In addition, the limits of integration for double integrals are typically expressed as two variables instead of one.

4. What are some applications of multivariable calculus?

Multivariable calculus has many applications in various fields, including physics, engineering, economics, and statistics. It is used to study and model complex systems, such as fluid flow, electric fields, and economic markets. It also plays a crucial role in optimization problems and in understanding the behavior of functions with multiple variables.

5. What are some common techniques used in multivariable calculus?

Some common techniques used in multivariable calculus include partial derivatives, multiple integrals, vector calculus, and the gradient, divergence, and curl operators. These techniques are used to analyze functions with multiple variables, calculate rates of change, and solve optimization problems.

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