Solving Double Integrals: Order of Integration Explained

In summary, the conversation is about someone asking for help with solving a double integral by changing the order of integration. They also mention a possible error in the original problem. The person responding suggests sketching the region of integration and using horizontal strips to determine the bounds.
  • #1
Estelle
1
0
Hey guys,

need your help and hope someone takes the time:

I need to solve the double integral by changing the order of Integration.

View attachment 4966

I would really appreciate if you could illustrate the way of how to compute the solution.

Best
Estelle :)
 

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  • #2
Hello and welcome to MHB, Estelle! :D

I have moved this thread, since it is a problem more likely found in the 3rd semester of an elementary calculus course.

Are you certain the problem isn't actually:

\(\displaystyle \int_{-3}^1\int_{2x}^{3-x^2}xy\,dy\,dx\)?
 
  • #3
Assuming that what Mark has posted is correct, to reverse the order of integration you need to SKETCH the region of integration, and then figure out your bounds by using horizontal strips instead of vertical ones...
 

FAQ: Solving Double Integrals: Order of Integration Explained

What is the purpose of solving double integrals?

Solving double integrals allows us to calculate the volume under a surface in a 3D space. It is also useful in finding the area between two curves or the mass of an object with varying density.

How do you determine the order of integration for a double integral?

The order of integration is determined by the limits of the inner integral. It should correspond to the variable that is being integrated first. The outer limits are then determined by the remaining variable.

What is the difference between changing the order of integration and switching the limits of integration?

Changing the order of integration means rearranging the order in which the variables are integrated. This may result in a simpler integral to solve. Switching the limits of integration, on the other hand, means swapping the upper and lower limits for a specific variable in the integral.

Can the order of integration be changed for any double integral?

Yes, the order of integration can be changed for any double integral as long as the limits of integration are not dependent on the order in which the variables are integrated.

How do you know if you have solved a double integral correctly?

To ensure that a double integral has been solved correctly, you should check that the final answer matches the given limits of integration and the function being integrated. You can also verify your solution by using a graphing calculator or software to visualize the surface under the integral.

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