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Question-121704




Question Number 121704 by rs4089 last updated on 11/Nov/20
Answered by bemath last updated on 11/Nov/20
 ∫_1 ^3 ∣ ( (1/2)x^2 y^2 )∣_1 ^2  dy = ∫_1 ^3 (2y^2 −(1/2)y^2 ) dy   = ∣(3/2).(1/3)y^3  ∣_1 ^3 = (1/2)(27−1) = 13.
$$\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\mid\:\left(\:\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)\mid_{\mathrm{1}} ^{\mathrm{2}} \:{dy}\:=\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\left(\mathrm{2}{y}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}{y}^{\mathrm{2}} \right)\:{dy} \\ $$$$\:=\:\mid\frac{\mathrm{3}}{\mathrm{2}}.\frac{\mathrm{1}}{\mathrm{3}}{y}^{\mathrm{3}} \:\mid_{\mathrm{1}} ^{\mathrm{3}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{27}−\mathrm{1}\right)\:=\:\mathrm{13}.\: \\ $$
Commented by rs4089 last updated on 11/Nov/20
limit 1 to 2 is limit of x, is not y , why ?
$${limit}\:\mathrm{1}\:{to}\:\mathrm{2}\:{is}\:{limit}\:{of}\:{x},\:{is}\:{not}\:{y}\:,\:{why}\:? \\ $$
Commented by bemath last updated on 11/Nov/20
because ∫_1 ^3  (∫_1 ^2  xy^2  dx ) dy
$${because}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\left(\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:{xy}^{\mathrm{2}} \:{dx}\:\right)\:{dy}\: \\ $$

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