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Question Number 34293 Answers: 0 Comments: 0

calculate ∫∫_D x^2 y dxdy? with D = {(x,y)∈ R^2 / 0≤y≤1−x^2 ,∣x+y +3∣ ≤5}

$${calculate}\:\int\int_{{D}} \:\:{x}^{\mathrm{2}} {y}\:{dxdy}?\:\:{with} \\ $$$${D}\:=\:\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}−{x}^{\mathrm{2}} \:,\mid{x}+{y}\:+\mathrm{3}\mid\:\leqslant\mathrm{5}\right\} \\ $$

Question Number 34292 Answers: 0 Comments: 1

calculate ∫∫_w (x+y)e^(x−y) dxdy with w={(x,y)∈R^2 / ∣x∣ ≤1 and ∣y+1∣≤3 }

$${calculate}\:\int\int_{{w}} \:\left({x}+{y}\right){e}^{{x}−{y}} {dxdy}\:{with} \\ $$$${w}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\mid{x}\mid\:\leqslant\mathrm{1}\:\:{and}\:\mid{y}+\mathrm{1}\mid\leqslant\mathrm{3}\:\right\} \\ $$

Question Number 34291 Answers: 0 Comments: 0

let B(x,y) = ∫_0 ^1 u^(x−1) (1−u)^(y−1) du and Γ(x)= ∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x) = 2∫_0 ^∞ u^(2x−1) e^(−u^2 ) du 2)give Γ(x)Γ(y) at form of double integrale 3)prove that B(x,y) =((Γ(x)Γ(y))/(Γ(x+y))) 4) calculate B(m,n) for m and n integr naturals

$${let}\:{B}\left({x},{y}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{y}−\mathrm{1}} \:{du}\:\:{and} \\ $$$$\Gamma\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{u}^{\mathrm{2}{x}−\mathrm{1}} \:{e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$\left.\mathrm{2}\right){give}\:\Gamma\left({x}\right)\Gamma\left({y}\right)\:{at}\:{form}\:{of}\:{double}\:{integrale} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{B}\left({m},{n}\right)\:{for}\:{m}\:{and}\:{n}\:{integr}\:{naturals} \\ $$

Question Number 34290 Answers: 0 Comments: 0

calculate ∫∫_D ((dxdy)/((1+x+y)^2 )) D ={(x,y)∈ R^2 / 1≤x+y≤ 2}

$${calculate}\:\int\int_{{D}} \:\:\:\:\:\:\:\frac{{dxdy}}{\left(\mathrm{1}+{x}+{y}\right)^{\mathrm{2}} } \\ $$$${D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:\:\mathrm{1}\leqslant{x}+{y}\leqslant\:\mathrm{2}\right\}\: \\ $$

Question Number 34289 Answers: 0 Comments: 1

calculate ∫∫_w (xy −2)dxdy with w = {(x,y)∈R^2 / x≥0 and 1≤y≤2−x }

$$\:{calculate}\:\int\int_{{w}} \:\:\left({xy}\:−\mathrm{2}\right){dxdy}\:\:{with}\: \\ $$$${w}\:=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\:{x}\geqslant\mathrm{0}\:{and}\:\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}−{x}\:\right\} \\ $$

Question Number 34288 Answers: 0 Comments: 0

calculate ∫∫_w e^(−yx^2 ) (x+y)dxdy with w =[0,1]^2

$${calculate}\:\int\int_{{w}} \:{e}^{−{yx}^{\mathrm{2}} } \left({x}+{y}\right){dxdy}\:\:{with} \\ $$$${w}\:=\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} \\ $$

Question Number 34287 Answers: 0 Comments: 0

calculate ∫∫_D xydxdy with D={(x,y)∈R^2 /x≥0 ,y≥0 , x+y ≤ (3/2)}