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Question Number 144756 by mondlihk last updated on 28/Jun/21

Commented by Mathspace last updated on 28/Jun/21

	Answered by Olaf_Thorendsen last updated on 28/Jun/21
	$Ω = ∫∫_(D = {x≥0, y≥0, x+y≤1}) xy dxdy Ω = ∫_0 ^1 ∫_0 ^(1−x) xy dxdy Ω = ∫_0 ^1 [x(y^2 /2)]_0 ^(1−x) dx Ω = ∫_0 ^1 x(((1−x)^2 )/2) dx Ω = [(x^4 /8)−(x^3 /3)+(x^2 /4)]_0 ^1 = ((3−8+6)/(24)) = (1/(24))$
	$Ω = \int \int_{D = {x ⩾ 0, y ⩾ 0, x + y ⩽ 1}} x y d x d y$ $Ω = \int_{0}^{1} \int_{0}^{1 - x} x y d x d y$ $Ω = \int_{0}^{1} {[x \frac{y^{2}}{2}]}_{0}^{1 - x} d x$ $Ω = \int_{0}^{1} x \frac{{(1 - x)}^{2}}{2} d x$ $Ω = {[\frac{x^{4}}{8} - \frac{x^{3}}{3} + \frac{x^{2}}{4}]}_{0}^{1} = \frac{3 - 8 + 6}{24} = \frac{1}{24}$
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