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Category: Integration

please-help-me-to-find-this-a-D-ydxdy-a-2-x-2-D-x-0-y-0-x-2-y-2-a-2-b-v-x-y-z-2-dxdydz-v-x-0-y-0-z-0-x-z-1-y-z-1-c-V-xydxdydz-V-0-z-1-x-2-y-2-

Question Number 170545 by kndramaths last updated on 26/May/22 $$\:\:\:\:{please}\:{help}\:{me}\:{to}\:{find}\:{this}. \\ $$$$\:\:\:{a}=\:\int\int_{{D}} \frac{{ydxdy}}{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }\:{D}:\left\{{x}\geqslant\mathrm{0}.{y}\geqslant\mathrm{0}.{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \leqslant{a}^{\mathrm{2}} \right\} \\ $$$$\:\:\:{b}=\int\int\int_{{v}} \left({x}−{y}+{z}\right)^{\mathrm{2}} {dxdydz} \\ $$$$\:\:{v}:\left\{{x}=\mathrm{0}.{y}=\mathrm{0}.{z}=\mathrm{0}\:{x}+{z}=\mathrm{1}.{y}+{z}=\mathrm{1}\right\} \\…

c-cosxsiny-xy-dx-sinx-cosy-dy-faind-integral-on-the-opposite-sid-of-the-clock-face-in-the-c-unit-circle-solve-this-

Question Number 170535 by mathlove last updated on 26/May/22 $$\int_{{c}} \left({cosxsiny}−{xy}\right){dx}+\left({sinx}\:\centerdot{cosy}\right){dy} \\ $$$${faind}\:{integral}\:{on}\:{the}\:{opposite} \\ $$$${sid}\:{of}\:{the}\:{clock}\:{face}\:{in}\:{the} \\ $$$${c}\:{unit}\:{circle}? \\ $$$${solve}\:{this} \\ $$ Commented by mathlove last…

Let-I-n-x-n-e-x-dx-n-0-1-2-i-Show-that-I-n-x-n-e-x-nI-n-1-ii-Show-that-0-x-n-e-x-dx-n-

Question Number 170532 by MikeH last updated on 26/May/22 $$\mathrm{Let}\:{I}_{{n}} \:=\int{x}^{{n}} {e}^{−{x}} {dx},\:{n}\:=\:\mathrm{0},\mathrm{1},\mathrm{2},… \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:{I}_{{n}} \:=\:−{x}^{{n}} {e}^{−{x}} +{nI}_{{n}−\mathrm{1}} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} {x}^{{n}} {e}^{−{x}} {dx}\:=\:{n}! \\…