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
![let I =∫_0 ^(100π) ∣sinx∣dx ⇒I =Σ_(k=0) ^(99) ∫_(kπ) ^((k+1)π) ∣sinx∣dx =_(x=kπ +u) Σ_(k=0) ^(99) ∫_0 ^π ∣(−1)^k sinu∣ du =Σ_(k=0) ^(99) ∫_0 ^π sinu du =Σ_(k=0) ^(99) [−cosu]_0 ^π =2 Σ_(k=0) ^(99) (1) =2×100 =200](https://www.tinkutara.com/question/Q117016.png)
$$\mathrm{let}\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid\mathrm{sinx}\mid\mathrm{dx}\:\:\Rightarrow\mathrm{I}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\int_{\mathrm{k}\pi} ^{\left(\mathrm{k}+\mathrm{1}\right)\pi} \:\:\mid\mathrm{sinx}\mid\mathrm{dx} \\ $$$$=_{\mathrm{x}=\mathrm{k}\pi\:+\mathrm{u}} \:\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\:\int_{\mathrm{0}} ^{\pi} \mid\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{sinu}\mid\:\mathrm{du}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \:\int_{\mathrm{0}} ^{\pi} \mathrm{sinu}\:\mathrm{du} \\ $$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \left[−\mathrm{cosu}\right]_{\mathrm{0}} ^{\pi} \:\:=\mathrm{2}\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{99}} \left(\mathrm{1}\right)\:=\mathrm{2}×\mathrm{100}\:=\mathrm{200} \\ $$
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
$$\mid{sinx}\mid\:{graph}\:{is}\:{alaways}\:+{ve}\:{and}\:{each}\:{loop}\:{of}\: \\ $$$${sinx}\:{area}\:{is}\:\mathrm{2}.\:{here}\:\mathrm{100}\:{loop}\:{so}\:{area}\:{is}\:\mathrm{2}×\mathrm{100}=\mathrm{200} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid{sinx}\mid{dx}=\mathrm{100}\int_{\mathrm{0}} ^{\pi} \mid{sinx}\mid{dx}=\mathrm{100}\mid−{cosx}\mid_{\mathrm{0}} ^{\pi} \\ $$$$=−\mathrm{100}\left(−\mathrm{1}−\mathrm{1}\right)=\mathrm{200} \\ $$$$ \\ $$