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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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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