Question Number 79499 by jagoll last updated on 25/Jan/20
$$\underset{\mathrm{0}} {\overset{\mathrm{30}\pi} {\int}}\mid\mathrm{sin}\:\mathrm{x}\mid\:\mathrm{dx}=\: \\ $$
Commented by john santu last updated on 25/Jan/20
![y = ∣sin x∣ is even function and periodic with periode = π ∫_0 ^(30π) ∣sin x∣dx = ∫_0 ^π ∣sin x∣dx+∫_π ^(2π) ∣sin x∣dx+ ...+∫_(29π) ^(30) ∣sin x∣dx = 30×[2∫_0 ^(π/2) sin xdx] = 60×1 = 60.](Q79501.png)
$${y}\:=\:\mid\mathrm{sin}\:{x}\mid\:{is}\:{even}\:{function}\:{and} \\ $$$${periodic}\:{with}\:{periode}\:=\:\pi \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{30}\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}\:=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}+\underset{\pi} {\overset{\mathrm{2}\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}+ \\ $$$$...+\underset{\mathrm{29}\pi} {\overset{\mathrm{30}} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}\: \\ $$$$=\:\mathrm{30}×\left[\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{sin}\:{xdx}\right]\:=\:\mathrm{60}×\mathrm{1}\:=\:\mathrm{60}. \\ $$
Commented by mathmax by abdo last updated on 25/Jan/20
![∫_0 ^(30π) ∣sinx∣dx =Σ_(k=0) ^(29) ∫_(kπ) ^((k+1)π) ∣sinx∣dx =_(x=kπ +t) =Σ_(k=0) ^(29) ∫_0 ^π ∣sin(kπ +t)dt =Σ_(k=0) ^(29) ∫_0 ^π ∣sint∣dt =Σ_(k=0) ^(29) ∫_0 ^π sint dt =Σ_(k=0) ^(29) [−cost]_0 ^π =2Σ_(k=0) ^(29) (1) =2×30 =60](Q79523.png)
$$\int_{\mathrm{0}} ^{\mathrm{30}\pi} \mid{sinx}\mid{dx}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\:\int_{{k}\pi} ^{\left({k}+\mathrm{1}\right)\pi} \mid{sinx}\mid{dx}\:=_{{x}={k}\pi\:+{t}} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} \mid{sin}\left({k}\pi\:+{t}\right){dt}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} \mid{sint}\mid{dt} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} {sint}\:{dt}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \left[−{cost}\right]_{\mathrm{0}} ^{\pi} \:=\mathrm{2}\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \left(\mathrm{1}\right)\:=\mathrm{2}×\mathrm{30}\:=\mathrm{60} \\ $$