Sketch Region of Integration, Reverse Order, Confirm Equality

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SUMMARY

The discussion focuses on reversing the order of integration for the double integral \(\int\int dydx\) over the region defined by \(0 \leq x \leq 1\) and \(0 \leq y \leq \sqrt{x}\). The user initially attempted to rewrite the integral as \(\int\int dxdy\) with limits \(0 \leq x \leq y^2\) and \(0 \leq y \leq 1\), but encountered discrepancies in the evaluated results, yielding \(2/3\) and \(1/3\) instead of the expected \(1/6\). The discussion emphasizes the importance of correctly sketching the region of integration and adjusting the limits accordingly when changing the order of integration.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with changing the order of integration
  • Ability to sketch regions of integration
  • Knowledge of evaluating definite integrals
NEXT STEPS
  • Review the process of sketching regions of integration in double integrals
  • Learn about the correct limits of integration when reversing order
  • Practice evaluating double integrals with various limits
  • Study examples of common mistakes in changing the order of integration
USEFUL FOR

Students studying calculus, particularly those focusing on multivariable calculus and double integrals, as well as educators looking for examples of common pitfalls in integration techniques.

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Homework Statement



sketch the region of integration and write an equivalent integral with order of integration reversed. Then evaluate both integrals to confirm their equality

Homework Equations



\int\intdydx for 0<=x<=1 and 0<=y<=\sqrt{x}

The Attempt at a Solution



i rearanged the limits so the equation becomes
\int\intdxdy for 0<=x<=y2 and 0<=y<=1

but my calculations for the first equation came to 2/3 and the second equation came to 1/3

plus the answer is 1/6 so I am obviously doing something wrong. can someone help? have i even written the second equation right?
 
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Did you first sketch the region of integration? The limits for the second integral(with order of integration reversed) should be taken such that you still integrate over the same region. Do you know how to do this?
 
yeh I am pretty sure i drew it properly??
 

Attachments

Since your attachment is pending approval, assuming your diagram is correct, think about the new set of limits you have to use when you change the order of integration. How do x and y vary now?
 

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