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evaluate ∫ ∫ R f(x,y) dy dx over the region

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1. R= {(x,y)| 0 ≤ x ≤ 1 and 0 ≤ y ≤ x} where f(x,y)= 24y^2e^(x^4+1)

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2. R={(x,y)| 0 ≤ x ≤ y and 0 ≤ y ≤ 2 } where f(x,y) = 12e^(4y+2)

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Fubinis Theorem:

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If $\"\"$ is continuous on the rectangle $\"\"$ then $\"\"$ .

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(1)

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$\"\"$ and $\"\"$ .

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Evaluate $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Fubinis Theorem:

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If $\"\"$ is continuous on the rectangle $\"\"$ then $\"\"$ .

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(1)

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$\"\"$ and $\"\"$ .

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Evaluate $\"\"$ .

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$\"\"$

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$\"\"$

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Apply integration by parts.

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Integration by parts: $\"\"$ .

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Consider $\"\"$ then $\"\"$ .

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Consider $\"\"$ then $\"\"$ .

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$\"\"$

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$\"\"$

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3. Find the volume of the solid under the graph of the function f(x,y) = 18x^2y^2 and over the region R= {(x,y)| 0 ≤ x ≤ 1 and -1 ≤ y ≤ 2 }

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(3)

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Find the volume of the solid.

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$\"\"$ and $\"\"$ .

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Evaluate $\"\"$ .

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$\"\"$

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$\"\"$