Answered: Which order of integration is…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.2: Applications Of Extrema

Problem 1YT: Find two nonnegative number x and y for which x+3y=30, such that x2y is maximized.

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Question

Which order of integration is preferable to integrate ƒ(x, y) = xy
over R = {(x, y): y - 1 ≤ x ≤ 1 - y, 0 ≤ y ≤ 1}?

Expert Solution

Step 1

Given:

$f (x, y) = x y R = \{(x, y) : y - 1 \leq x \leq 1 - y, 0 \leq y \leq 1\}$

To Determine:

Which order of integration is Preferable to integrate.

Step 2

Explanation:

$f (x, y) = x y R = \{(x, y) : y - 1 \leq x \leq 1 - y, 0 \leq y \leq 1\} Case - 1 (d x d y) : I_{1} = \int_{0}^{1} \int_{y - 1}^{1 - y} (x y) d x d y = \int_{0}^{1} y {[\frac{x^{2}}{2}]}_{y - 1}^{1 - y} d y = \frac{1}{2} \int_{0}^{1} y [{(1 - y)}^{2} - {(y - 1)}^{2}] d y \Rightarrow I_{1} = \frac{1}{2} \int_{0}^{1} y [1 + y^{2} - 2 y - (y^{2} + 1 - 2 y)] d y = \frac{1}{2} \int_{0}^{1} y (0) d y = 0 \Rightarrow I_{1} = 0 . Case - 2 (d y d x) : I_{2} = \int_{y - 1}^{1 - y} \int_{0}^{1} (x y) d y d x = \int_{y - 1}^{1 - y} x {[\frac{y^{2}}{2}]}_{0}^{1} d y = \frac{1}{2} \int_{y - 1}^{1 - y} x d y = \frac{1}{2} {[\frac{x^{2}}{2}]}_{y - 1}^{1 - y} \Rightarrow I_{2} = \frac{1}{4} [(1 + y^{2} - 2 y) - (y^{2} + 1 - 2 y)] = \frac{1}{4} (0) \Rightarrow I_{2} = 0 .$

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