Double Integral over General Region : Hass Section 13.2 - Problem 5

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SUMMARY

The discussion centers on solving a double integral problem from Hass Section 13.2, Problem 5, involving the function 3y³ * e^(xy) with outer integral limits from 0 to 1 and inner integral limits from 0 to y². The user struggles with the integration process, particularly with the inner integral, and questions the transition from 3y³ to 3y² as shown in the solutions manual. The correct approach involves recognizing the need for integration by parts to simplify the integration of the exponential function.

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  • Knowledge of exponential functions and their properties
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  • Practice evaluating double integrals with varying limits
  • Explore the properties of exponential functions in integration
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Students studying calculus, particularly those focusing on multivariable calculus and integration techniques, as well as educators looking for examples of double integrals and integration by parts.

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Homework Statement



Outer Integral: From zero to one dy
Inner Integral: from zero to y^2 dx

Function is: 3y^3 * e^(xy)


Homework Equations


None


The Attempt at a Solution



Have tried numerous u substitutions on e^(xy), but taking me nowhere. I am clearly doing something wrong. Assuming 3y^3 is a constant and does not need to be integrated when integrated with respect to x.

Solutions manual shows result of inner integral being [3y^2 * e^(xy)] from zero to y^2 - which appears to me that a y in the original 3y^3 simply disappeared! No idea how they are getting from 3y^3 to 3y^2 as the result of the first integration!

I am quite sure this is an easy problem and I am simply overlooking a very simple step.
 
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You need to use integration by parts.
 

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