Integration Questions

Question Number 117006 by TANMAY PANACEA last updated on 08/Oct/20

$$\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid{sinx}\mid\:{dx} \\$$

Commented by TANMAY PANACEA last updated on 08/Oct/20

$${thank}\:{you}\:{sir} \\$$$$\\$$

Answered by mathmax by abdo last updated on 08/Oct/20

$$\mathrm{let}\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid\mathrm{sinx}\mid\mathrm{dx}\:\:\Rightarrow\mathrm{I}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\int_{\mathrm{k}\pi} ^{\left(\mathrm{k}+\mathrm{1}\right)\pi} \:\:\mid\mathrm{sinx}\mid\mathrm{dx} \\$$$$=_{\mathrm{x}=\mathrm{k}\pi\:+\mathrm{u}} \:\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\:\int_{\mathrm{0}} ^{\pi} \mid\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{sinu}\mid\:\mathrm{du}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\int_{\mathrm{0}} ^{\pi} \mathrm{sinu}\:\mathrm{du} \\$$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \left[−\mathrm{cosu}\right]_{\mathrm{0}} ^{\pi} \:\:=\mathrm{2}\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \left(\mathrm{1}\right)\:=\mathrm{2}×\mathrm{100}\:=\mathrm{200} \\$$

Commented by TANMAY PANACEA last updated on 08/Oct/20

$${thank}\:{you}\:{sir} \\$$

Commented by Bird last updated on 09/Oct/20

$${you}\:{are}\:{welcome}\:{sir} \\$$

Answered by TANMAY PANACEA last updated on 08/Oct/20

$$\mid{sinx}\mid\:{graph}\:{is}\:{alaways}\:+{ve}\:{and}\:{each}\:{loop}\:{of}\: \\$$$${sinx}\:{area}\:{is}\:\mathrm{2}.\:{here}\:\mathrm{100}\:{loop}\:{so}\:{area}\:{is}\:\mathrm{2}×\mathrm{100}=\mathrm{200} \\$$$$\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid{sinx}\mid{dx}=\mathrm{100}\int_{\mathrm{0}} ^{\pi} \mid{sinx}\mid{dx}=\mathrm{100}\mid−{cosx}\mid_{\mathrm{0}} ^{\pi} \\$$$$=−\mathrm{100}\left(−\mathrm{1}−\mathrm{1}\right)=\mathrm{200} \\$$$$\\$$