GATE EC 2017 Set 2 — Question 26
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Engineering Mathematics → Calculus → Double & Triple Integrals
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Question
The values of the integralsandare
Correct answer
(C) 0.5 and - 0.5, respectively
Solution
Let the first integral be and the second integral be .The integrand is not absolutely integrable over the square because it has a singularity at and changes sign. Therefore, Fubini's theorem does not apply, and the order of integration matters.For :
First, evaluate the inner integral with respect to :Let , so . Then , and .
When , . When , .Now, evaluate the outer integral for :For :
First, evaluate the inner integral with respect to :Let , so . Then , and .
When , . When , .Now, evaluate the outer integral for :So, the values of the integrals are and , respectively.The final answer is .
First, evaluate the inner integral with respect to :Let , so . Then , and .
When , . When , .Now, evaluate the outer integral for :For :
First, evaluate the inner integral with respect to :Let , so . Then , and .
When , . When , .Now, evaluate the outer integral for :So, the values of the integrals are and , respectively.The final answer is .
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