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Question Number 20202 by Joel577 last updated on 24/Aug/17

$$\mathrm{Is}\mathrm{definite}\mathrm{integral}\mathrm{can}\mathrm{have}\mathrm{negative}\mathrm{value}?$$$$\mathrm{Because}\mathrm{I}\mathrm{think}{\int }_a^{b}f\left(x\right)dx\mathrm{is}\mathrm{total}\mathrm{area}\mathrm{below}$$$$\mathrm{graph}f\left(x\right)\mathrm{from}x=a\mathrm{until}x=b,\mathrm{so}\mathrm{it}{\mathrm{can}}^{\prime }\mathrm{t}$$$$\mathrm{be}\mathrm{negative}$$

Commented by ajfour last updated on 24/Aug/17

$$cannotf\left(x\right)itselfbenegativefor$$$$a⩽x⩽b?$$$$So{\int }_a^{b}f\left(x\right)dxcanbenegative,$$$$positiveorzero,dependingon$$$$valuesoff\left(x\right)inthelengthx=a$$$$tox=b.$$

Commented by Joel577 last updated on 24/Aug/17

$$\mathrm{In}\mathrm{example}I={\int }_0^{\pi /2}x\cos 2xdx$$$$\mathrm{the}\mathrm{result}\mathrm{is}-\frac{1}{2}.\mathrm{Is}\mathrm{it}\mathrm{true}\mathrm{or}\mathrm{not}?$$

Commented by ajfour last updated on 24/Aug/17

$${\int }_{\pi }^{2\pi }\sin xdx=?,{\int }_{-1}^{0}xdx=?.$$$$\int x\cos 2xdx=\frac{x\sin 2x}{2}{\mid }_0^{\pi /2}-\frac{1}{2}{\int }_0^{\pi /2}\sin 2xdx$$$$=0+\frac{\cos 2x}{4}{\mid }_0^{\pi /2}=\frac{-1-1}{4}=-\frac{1}{2}.$$

Commented by Joel577 last updated on 24/Aug/17

$$\mathrm{But}\mathrm{if}\mathrm{I}\mathrm{separate}\mathrm{it}\mathrm{into}$$$${\int }_0^{\pi /4}x.\cos 2xdx+{\int }_{\pi /4}^{\pi /2}x.\cos 2xdx$$$$\mathrm{it}{\mathrm{isn}}^{\prime }\mathrm{t}-\frac{1}{2}$$

Commented by ajfour last updated on 24/Aug/17

$$[\frac{x\sin 2x}{2}+\frac{\cos 2x}{4}]{\mid }_0^{\pi /4}+[\frac{x\sin 2x}{2}+\frac{\cos 2x}{4}]{\mid }_{\pi /4}^{\pi /2}$$$$=(\frac{\pi }{8}-\frac{1}{4})+(-\frac{1}{4}-\frac{\pi }{8})=-\frac{1}{2}.$$

Commented by Joel577 last updated on 24/Aug/17

Commented by Joel577 last updated on 24/Aug/17

$$\mathrm{I}\mathrm{have}\mathrm{a}\mathrm{little}\mathrm{doubt}\mathrm{about}\mathrm{this}$$$$\mathrm{The}\mathrm{area}\mathrm{between}\frac{\pi }{4}\mathrm{and}\frac{\pi }{2}\mathrm{is}\mathrm{below}\mathrm{x}-\mathrm{axis}$$$$\mathrm{So},\mathrm{the}\mathrm{integral}\mathrm{must}\mathrm{be}\mathrm{positive},{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{it}?$$

Commented by Joel577 last updated on 24/Aug/17

$$\mathrm{Please}\mathrm{explain}\mathrm{with}\mathrm{more}\mathrm{detail}\mathrm{Sir}\mathrm{Ajfour}$$

Commented by ajfour last updated on 24/Aug/17

$$Theareafrom\pi /4to3\pi /4is$$$$negative.$$

Commented by Joel577 last updated on 25/Aug/17

$$Understood.Thankyouverymuch$$

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