Integrating over a genral region (multivariable)

In summary, for integrating over type 1 regions, the bounds for the inner integral must be the functions of x that define the boundary of the region, rather than the outer integral, in order for the result to be a number rather than a function.
  • #1
maccha
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For integrating over type 1 and type 2 regions, why does the g(x) or g(y) bound have to be the inner integral? Thanks!
 
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  • #2
I have no idea what you are asking! What in the world is "g"? Where did it come from?

I think you mean this: suppose you are to integrate
[tex]\int_R\int f(x,y)dy[/tex]
where "R" is a "type 1 region". That means that there exist some numbers, x0 and x1, such that the region's boundary can be written as two separate functions of x, y= g1(x) and y= g2(x), such that g1(x)> g2(x) for all x between x0 and x1. That allow us to treat it as an integral over the area between g1(x) and g2 so that those are the bounds on the integral.

If you are asking why they must be the bounds on the inner integral rather than the outer, the answer is simply that the double integral result must be a number, not a function either x or y. If the bounds on the outer integral were functions of x or y, then so would the result of the integration be.
 

FAQ: Integrating over a genral region (multivariable)

What is the purpose of integrating over a general region in multivariable calculus?

Integrating over a general region in multivariable calculus allows us to find the total value or amount of a function over a specific area or volume. It is an essential tool in applications such as finding the center of mass, calculating work done, and determining total charge or probability.

How is integrating over a general region different from integrating over a single variable?

Integrating over a general region involves taking into account multiple variables, such as x, y, and z, whereas integrating over a single variable only considers one variable. Additionally, integrating over a general region requires the use of multiple integration techniques, such as double or triple integrals, while integrating over a single variable only requires a single integral.

What are some common methods for integrating over a general region?

Some common methods for integrating over a general region include using rectangular or polar coordinates, using the change of variables formula, and using the divergence theorem or Stokes' theorem to convert the integral into a simpler form. Different methods may be more suitable for different types of regions and functions.

How does the choice of the region affect the process of integration?

The choice of the region can greatly affect the process of integration. For example, integrating over a rectangular region is typically simpler than integrating over a more complex, irregularly shaped region. The choice of region also determines the limits of integration and can affect the choice of integration method.

What are some real-world applications of integrating over a general region?

Integrating over a general region has many real-world applications, such as calculating the mass or volume of an object with varying density, finding the average temperature of a room, determining the probability of an event occurring in a given area, and calculating the total force or work done by a variable force field.

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