How to evaluate this double integral?

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SUMMARY

The discussion focuses on evaluating the double integral ∫0 to 2 ∫x/4 to 1/2 (sin (πy²)) dy dx. Participants suggest converting the integral to polar coordinates or using a change of variables to simplify the evaluation. A key recommendation is to sketch the region of integration in the xy-plane to facilitate switching the order of integration, which may lead to a more manageable form of the integral. The substitution rule is also highlighted as a potential method for solving the integral.

PREREQUISITES
  • Understanding of double integrals and their evaluation
  • Familiarity with polar coordinates and coordinate transformations
  • Knowledge of trigonometric functions, specifically sin(πy²)
  • Ability to sketch regions in the xy-plane for integration
NEXT STEPS
  • Learn about changing variables in double integrals
  • Study the process of switching the order of integration
  • Explore the use of polar coordinates in multivariable calculus
  • Investigate the substitution rule for integrals involving trigonometric functions
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable integration techniques and methods for evaluating complex integrals.

Kuma
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Homework Statement



∫0 to 2 ∫x/4 to 1/2 (sin (pi*y2)) dy dx

Homework Equations





The Attempt at a Solution



I think I have to convert this to polar or do some sort of change of variable.

Although in polar y = r sin θ, so then you would have sin of a sin??
 
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Try drawing the region and switching the order of integration, and this might allow you to use substitution rule hopefully, as I believe you will get something of the form \int y*\sin(\pi y^2) dy
 
Kuma said:

Homework Statement



∫0 to 2 ∫x/4 to 1/2 (sin (pi*y2)) dy dx

Homework Equations





The Attempt at a Solution



I think I have to convert this to polar or do some sort of change of variable.

Although in polar y = r sin θ, so then you would have sin of a sin??

Sketch the region (in the xy-plane) over which the integration is being done. Use this to switch the order of integration.
 

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