Find total charge (using double integration)

In summary, the question is asking about the difference in results when setting up a double integration for finding total charge in a region, with x having a lower bound of 0 and upper bound of 5, and y having a lower bound of 2 and upper bound of 5. It is based on the knowledge of double integration and how the order of dx and dy can affect the results. Without further information, it is impossible to determine why the results may differ from the solution in the textbook. However, if the region of integration is a rectangle, the order of integration does not make a difference in the final answer.
  • #1
Max
9
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The question asks to find total charge in a region given x has lower bound 0 - upper bound 5 , y has lower bound as 2 and upper bound as 5. Based on knowledge I have been reading throughout the chapter, I set up a double integration with those dxdy, but the results went out to be off - compared to the solution textbook that set up their double integration the other way around which was dydx.

My question is whether the order of dy and dx for double integration makes the difference when you attempt to find the volume?
Thanks!
 
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  • #2
Max said:
The question asks to find total charge in a region given x has lower bound 0 - upper bound 5 , y has lower bound as 2 and upper bound as 5. Based on knowledge I have been reading throughout the chapter, I set up a double integration with those dxdy, but the results went out to be off - compared to the solution textbook that set up their double integration the other way around which was dydx.

My question is whether the order of dy and dx for double integration makes the difference when you attempt to find the volume?
Thanks!
If the region of integration is the rectangle ##\{(x, y) | 0 \le x \le 5, 2 \le y \le 5\}##, you should be able to integrate in either order. If the region is not a rectangle, then changing the order of integration makes a difference as far as the limits of integration go.

Since you haven't provided further information about the problem and your work, it's impossible to say why your answer disagrees with the answer in the book.
 
  • #3
Ah, thank you!
 
  • #4
And yes, today in lecture, the professor answered the same way as the region was a rectangle, so the boundaries inside should be in terms of the other variables while the boundary outside should be numbers
 
  • #5
Max said:
And yes, today in lecture, the professor answered the same way as the region was a rectangle, so the boundaries inside should be in terms of the other variables while the boundary outside should be numbers
This makes no sense. If the region of integration is a rectangle, your integral will look something like this:
##\int_{x = 0}^5 \int_{y = 2}^5~f(x, y)dy~dx##.
Switching the order of integration gives ##\int_{y = 2}^5 \int_{x = 0}^5~f(x, y)dx~dy##.
In both cases, the limits of integration don't involve variable expressions. I include "x = ..." and "y = ..." in the lower limits of integration only to show the name of the variable.
 
  • #6
Will you please specify the question?
 
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What is the concept of finding total charge using double integration?

The concept of finding total charge using double integration is a mathematical method that allows you to determine the total charge within a given region by integrating the charge density function over the entire volume of the region. This method is commonly used in physics and engineering to calculate the electric charge of a system.

What is the formula for calculating total charge using double integration?

The formula for calculating total charge using double integration is Q = ∬ρ(x,y,z) dV, where Q is the total charge, ρ is the charge density function, and dV is the differential volume element. This formula is derived from the principle of superposition, which states that the total charge is equal to the sum of all individual charges within a given region.

What are the steps involved in finding total charge using double integration?

The steps involved in finding total charge using double integration are as follows:

  1. Identify the region in which the charge is located.
  2. Write the charge density function ρ(x,y,z) in terms of the variables x, y, and z.
  3. Set up the double integral Q = ∬ρ(x,y,z) dV, with the limits of integration being the boundaries of the region in each variable.
  4. Evaluate the double integral using appropriate integration techniques.

Are there any limitations to using double integration to find total charge?

Yes, there are some limitations to using double integration to find total charge. One limitation is that the region must have a defined boundary, otherwise the integration cannot be performed. Additionally, the charge density function must be known or able to be determined for the entire region. In some cases, the integration may be very complex and difficult to evaluate, making it impractical to use double integration.

What are some real-life applications of finding total charge using double integration?

Double integration to find total charge has many real-life applications, such as calculating the electric charge of a capacitor, determining the electric field of a charged particle, and analyzing the charge distribution in a complex system. It is also used in fields such as electromagnetism, circuit design, and fluid dynamics to understand the behavior of electric charges in various systems.

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