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What is the double integral of:f(x,y)=e^(x+y) when R is the area bounded by y=x+1,...

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bhanbhan | Student, Undergraduate | eNoter

Posted March 10, 2012 at 3:18 AM via web

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What is the double integral of:
f(x,y)=e^(x+y) when R is the area bounded by y=x+1, y=x-1, y=1-x, y=-1-x?

How to find R?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 11, 2012 at 12:04 AM (Answer #1)

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You need to solve for y the inner integral, considering x as constant, such that:

`int_(x-1)^(x+1) e^(x+y) dy = e^(x+y)|_(x-1)^(x+1)`

`int_(x-1)^(x+1) e^(x+y) dy = e^(x+x+1) - e^(x+x-1)`

`int_(x-1)^(x+1) e^(x+y) dy = e^(2x+1) - e^(2x-1)`

You need to solve for x the outer integral such that:

`int_(-1-y)^(1-y) e^(2x+1) dx - int_(-1-y)^(1-y)e^(2x-1) dx = (1/2)(e^(2x+1) - e^(2x-1))|_(-1-y)^(1-y)`

`int_(-1-y)^(1-y) e^(2x+1) dx - int_(-1-y)^(1-y)e^(2x-1) dx = (1/2)(e^(2-2y+1) - e^(-2 - 2y+1) - e^(2 - 2y -1) + e^(-2 - 2y - 1))`

`int_(-1-y)^(1-y) e^(2x+1) dx - int_(-1-y)^(1-y)e^(2x-1) dx = (1/2)(e^(3-2y) - e^(-1 - 2y) - e^(1 - 2y) + e^(-3 - 2y))`

 Hence, evaluating the double integral under given conditions yields

`int_(-1-y)^(1-y) int_(x-1)^(x+1) e^(x+y) dy dx = (1/2)(e^(3-2y) - e^(-1 - 2y) - e^(1 - 2y) + e^(-3 - 2y)).`

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