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Question Number 79499 by jagoll last updated on 25/Jan/20

$$\underset{0}{\stackrel{30\pi }{\int }}\mid \sin \mathrm{x}\mid \mathrm{dx}=$$

Commented by john santu last updated on 25/Jan/20

$$y=\mid \sin x\mid isevenfunctionand$$$$periodicwithperiode=\pi$$$$\underset{0}{\stackrel{30\pi }{\int }}\mid \sin x\mid dx=\underset{0}{\stackrel{\pi }{\int }}\mid \sin x\mid dx+\underset{\pi }{\stackrel{2\pi }{\int }}\mid \sin x\mid dx+$$$$...+\underset{29\pi }{\stackrel{30}{\int }}\mid \sin x\mid dx$$$$=30\times \left[2\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\sin xdx\right]=60\times 1=60.$$

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

$${\int }_0^{30\pi }\mid sinx\mid dx=\sum _{k=0}^{29}{\int }_{k\pi }^{(k+1)\pi }\mid sinx\mid dx{=}_{x=k\pi +t}$$$$=\sum _{k=0}^{29}{\int }_0^{\pi }\mid sin(k\pi +t)dt=\sum _{k=0}^{29}{\int }_0^{\pi }\mid sint\mid dt$$$$=\sum _{k=0}^{29}{\int }_0^{\pi }sintdt=\sum _{k=0}^{29}{[-cost]}_0^{\pi }=2\sum _{k=0}^{29}\left(1\right)=2\times 30=60$$

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