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Category: Integration

0-a-f-x-dx-

Question Number 23432 by gopikrishnan005@gmail.com last updated on 30/Oct/17 $$\int_{\mathrm{0}} ^{{a}} {f}\left({x}\right){dx}= \\ $$ Answered by Joel577 last updated on 31/Oct/17 $$\mathrm{Let}\:{F}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{anti}\:\mathrm{derivative}\:\mathrm{of}\:{f}\left({x}\right) \\ $$$${I}\:=\:\underset{\mathrm{0}} {\overset{{a}}…

hello-floor-function-a-b-x-dx-a-b-z-and-b-gt-a-0-b-x-dx-0-a-x-dx-b-2-b-2-a-2-a-2-1-now-m-k-x-dx-when-m-k-z-when-m-lt-a-lt-b-lt-k-b-k-and-a

Question Number 88955 by M±th+et£s last updated on 14/Apr/20 $${hello}\: \\ $$$${floor}\:{function} \\ $$$$\int_{{a}} ^{{b}} \lfloor{x}\rfloor\:\:{dx}\:\:\:\:\:\:\:\:\:\:{a},{b}\in{z}\:\:\:{and}\:{b}>{a} \\ $$$$=\int_{\mathrm{0}} ^{{b}} \lfloor{x}\rfloor\:{dx}\:−\int_{\mathrm{0}} ^{{a}} \lfloor{x}\rfloor\:{dx}\:=\frac{{b}^{\mathrm{2}} −{b}}{\mathrm{2}}−\frac{{a}^{\mathrm{2}} −{a}}{\mathrm{2}}\:….\left(\mathrm{1}\right) \\…

find-A-0-cos-x-x-2-x-1-2-dx-with-gt-0-2-find-the-value-of-0-cos-3x-x-2-x-1-2-dx-

Question Number 88930 by mathmax by abdo last updated on 13/Apr/20 $${find}\:{A}_{\lambda} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\lambda{x}\right)}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\mathrm{3}{x}\right)}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$…

Show-that-volume-of-a-region-of-space-bounded-by-a-boundary-surface-S-is-V-1-3-S-rcos-dA-being-the-angle-between-the-position-vector-of-a-point-P-on-the-surface-and-the-outer-normal-to

Question Number 23393 by ajfour last updated on 29/Oct/17 $${Show}\:{that}\:{volume}\:{of}\:{a}\:{region} \\ $$$${of}\:{space}\:{bounded}\:{by}\:{a}\:{boundary} \\ $$$${surface}\:{S}\:{is}\:\:{V}=\:\frac{\mathrm{1}}{\mathrm{3}}\underset{{S}\:} {\int\int}{r}\mathrm{cos}\:\theta{dA}\:. \\ $$$$\theta\:{being}\:{the}\:{angle}\:{between}\:{the} \\ $$$${position}\:{vector}\:{of}\:{a}\:{point}\:{P}\:\:{on} \\ $$$${the}\:{surface},\:{and}\:{the}\:{outer}\:{normal} \\ $$$${to}\:{the}\:{surface}\:{at}\:{P}. \\ $$$${r}\:{is}\:{the}\:{distance}\:{of}\:{point}\:{P}\:{from}…