Figure out the double integral&nbsp;Rye−xydA,R=[0,2]×[0,3]

To make the calculation easier, you should first integrate with respect to x. In essence it doesn't matter but recognize that the function under the integral is derivative of&nbsp;−e−xy&nbsp;with respect to x, hence we solve one integral without too much computation:∫03∫02ye−xy&nbsp;dx&nbsp;&nbsp;dy&nbsp;=∫03−e−2y+1&nbsp;dy&nbsp;=52+e−62Result:52+e−62

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High School
Calculus and Analysis
Calculus 1
Integrals

Steven Smith

Answered question

2021-12-05

Figure out the double integral $R y e^{- x y} d A, R = [0, 2] \times [0, 3]$

Answer & Explanation

Unpled

Beginner2021-12-06Added 23 answers

To make the calculation easier, you should first integrate with respect to x. In essence it doesn't matter but recognize that the function under the integral is derivative of $- e^{- x y}$ with respect to x, hence we solve one integral without too much computation:
$\int_{0}^{3} \int_{0}^{2} y e^{- x y} d x d y = \int_{0}^{3} - e^{- 2 y} + 1 d y = \frac{5}{2} + \frac{e^{- 6}}{2}$
Result:
$\frac{5}{2} + \frac{e^{- 6}}{2}$

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