Limits Questions

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 byjohn santu last updated on 25/Jan/20

$${y}\:=\:\mid\mathrm{sin}\:{x}\mid\:{is}\:{even}\:{function}\:{and} \\$$ $${periodic}\:{with}\:{periode}\:=\:\pi \\$$ $$\underset{\mathrm{0}} {\overset{\mathrm{30}\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}\:=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}+\underset{\pi} {\overset{\mathrm{2}\pi} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}+ \\$$ $$...+\underset{\mathrm{29}\pi} {\overset{\mathrm{30}} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}\: \\$$ $$=\:\mathrm{30}×\left[\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{sin}\:{xdx}\right]\:=\:\mathrm{60}×\mathrm{1}\:=\:\mathrm{60}. \\$$

Commented bymathmax by abdo last updated on 25/Jan/20

$$\int_{\mathrm{0}} ^{\mathrm{30}\pi} \mid{sinx}\mid{dx}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\:\int_{{k}\pi} ^{\left({k}+\mathrm{1}\right)\pi} \mid{sinx}\mid{dx}\:=_{{x}={k}\pi\:+{t}} \\$$ $$=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} \mid{sin}\left({k}\pi\:+{t}\right){dt}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} \mid{sint}\mid{dt} \\$$ $$=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \:\int_{\mathrm{0}} ^{\pi} {sint}\:{dt}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \left[−{cost}\right]_{\mathrm{0}} ^{\pi} \:=\mathrm{2}\sum_{{k}=\mathrm{0}} ^{\mathrm{29}} \left(\mathrm{1}\right)\:=\mathrm{2}×\mathrm{30}\:=\mathrm{60} \\$$