Solving Iterated Integrals with x, y Boundaries

  • Thread starter stunner5000pt
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In summary, you need to integrate for the yellow triangle, which has boundaries at x=0, x=sqrt(pi), and y=0. If you change the order of integration, it would go from x=0 to x=y; and then from y=0 to y=sqrt(pi).
  • #1
stunner5000pt
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i need to solve this
[tex] \int_{0}^{\sqrt{\pi}} ( \int_{x}^{\sqrt{\pi}} sin y^2 dy) dx [/tex]

now i know i have to change the order of this
the integrand is bounded by the triangle from x = 0 to [itex] x= \sqrt{\pi} [/itex] here's where i am stuck
what is the boundary of the y?? is y bounded below by x=0 and above by x =1??

so what would the limits of integration change to?? (for the inside one from 0 to root pi?) and the outside one stays the same??

pelase help!
 
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  • #2
stunner5000pt said:
i need to solve this
[tex] \int_{0}^{\sqrt{\pi}} ( \int_{x}^{\sqrt{\pi}} sin y^2 dy) dx [/tex]

now i know i have to change the order of this ...


Look at the attached figure. You have to integrate for the yellow triangle, according to the boundaries of your original integral. If you change the order of integration, that is you integrate by x first, it would go from x=0 to x=y; and then by y which goes from y=0 to y= sqrt (pi).

ehild
 
Last edited:
  • #3
ehild said:
Look at the attached figure. You have to integrate for the yellow triangle, according to the boundaries of your original integral. If you change the order of integration, that is you integrate by x first, it would go from x=0 to x=y; and then by y which goes from y=0 to y= sqrt (pi).

ehild

how do you know that it is the upper triangle and not the lower triangle??
 
  • #4
Because in the original integral:

[tex] \int_{0}^{\sqrt{\pi}} ( \int_{x}^{\sqrt{\pi}} sin y^2 dy) dx [/tex]

Look at the limits for the dy integral, y goes from y=x to y=root pi. If it were the white triangle in the image, then y would be going from 0 to x.
 
  • #5
ehild is completely correct

marlon
 
  • #6
You may want to search mathworld for Fresnel' S(x) antiderivative...

Daniel.
 
Last edited:

Related to Solving Iterated Integrals with x, y Boundaries

1. How do I solve iterated integrals with x, y boundaries?

Solving iterated integrals with x, y boundaries involves following a specific set of steps. First, you must identify the inner and outer functions and their respective boundaries. Then, integrate the inner function with respect to its boundary while treating the outer function as a constant. Finally, integrate the resulting expression with respect to the outer boundary.

2. What is the purpose of iterated integrals with x, y boundaries?

The purpose of iterated integrals with x, y boundaries is to solve for the volume under a 3-dimensional surface or the area under a 2-dimensional surface. It is a useful tool in many areas of science, such as physics, engineering, and economics.

3. What are some common mistakes to avoid when solving iterated integrals with x, y boundaries?

Some common mistakes to avoid when solving iterated integrals with x, y boundaries include incorrectly identifying the inner and outer functions, forgetting to treat the outer function as a constant during integration, and using the wrong boundaries for each integration step. It is important to carefully check each step of the process to avoid these errors.

4. How can I check my solution for an iterated integral with x, y boundaries?

One way to check your solution for an iterated integral with x, y boundaries is to use a graphing calculator or software to graph the original function and the integrated function. The area under the integrated function should match the value you obtained for the iterated integral. You can also check your solution by plugging in different values for x and y within the given boundaries and comparing the results to the original function.

5. Are there any tips for simplifying the process of solving iterated integrals with x, y boundaries?

One tip for simplifying the process of solving iterated integrals with x, y boundaries is to use symmetry whenever possible. If the function and boundaries have symmetry, you can simplify the integrations by only solving for one quadrant or one half of the surface and then doubling the result. Another helpful tip is to carefully choose the order of integration, which can make the process more efficient and reduce the chances of making mistakes.

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