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Question Number 32775 by NECx last updated on 02/Apr/18

$$Findtheareaboundedbythe$$$$curvey=3x^2+2x-3,thexaxis$$$$andthelinex=5andx=2$$

Answered by MJS last updated on 02/Apr/18

$$1.\mathrm{check}\mathrm{the}\mathrm{zeros}$$$$x^2+\frac{2}{3}x-1=0$$$$x=-\frac{1}{3}\pm \sqrt{\frac{1}{9}+1}=-\frac{1}{3}\pm \frac{\sqrt{10}}{3}$$$$x_1\approx -1.39\mathrm{and}{x}_2\approx .72\mathrm{are}\mathrm{both}$$$$\mathrm{inside}[-5;2]\Rightarrow \mathrm{changes}\mathrm{of}\mathrm{sign}$$$$$$$$2.\mathrm{area}:$$$$x<x_1\Rightarrow f\left(x\right)>0$$$$x\in ]x_1;x_2[\Rightarrow f\left(x\right)<0$$$$x>x_2\Rightarrow f\left(x\right)>0$$$$\underset{-5}{\stackrel{x_1}{\int }}f\left(x\right)dx-\underset{x_1}{\stackrel{x_2}{\int }}f\left(x\right)dx+\underset{x_2}{\stackrel{2}{\int }}f\left(x\right)dx=$$$$=F\left(x\right)\underset{-5}{\stackrel{x_1}{\mid }}-F\left(x\right)\underset{x_1}{\stackrel{x_2}{\mid }}+F\left(x\right)\underset{x_2}{\stackrel{2}{\mid }}=$$$$[F\left(x\right)=x^3+x^2-3x]$$$$=91+\frac{80\sqrt{10}}{27}$$

Commented by MJS last updated on 02/Apr/18

$$\mathrm{no}\mathrm{problem},\mathrm{posted}\mathrm{a}\mathrm{new}\mathrm{answer}$$

Commented by NECx last updated on 02/Apr/18

$$I^{\prime }msosorry...Imadeatypoerror$$$$thelinesarex=-5andx=2.$$

Commented by Joel578 last updated on 03/Apr/18

$$\mathrm{You}\mathrm{just}\mathrm{set}\mathrm{the}\mathrm{integral}$$$$\mathrm{A}=\underset{-5}{\stackrel{\frac{-1-\sqrt{10}}{3}}{\int }}f\left(x\right)dx+\underset{\frac{-1+\sqrt{10}}{3}}{\stackrel{\frac{-1-\sqrt{10}}{3}}{\int }}f\left(x\right)dx+\underset{\frac{-1+\sqrt{10}}{3}}{\stackrel{2}{\int }}f\left(x\right)dx$$

Commented by MJS last updated on 02/Apr/18

$$\mathrm{the}\mathrm{middle}\mathrm{integral}\mathrm{is}\mathrm{negative},$$$$\mathrm{for}\mathrm{the}\mathrm{area}\mathrm{you}\mathrm{need}\int \mid f\left(x\right)\mid ,\mathrm{so}$$$${\mathrm{it}}^{\prime }\mathrm{s}\underset{-5}{\stackrel{-\frac{1}{3}-\frac{\sqrt{10}}{3}}{\int }}f\left(x\right)dx-\underset{-\frac{1}{3}-\frac{\sqrt{10}}{3}}{\stackrel{-\frac{1}{3}+\frac{\sqrt{10}}{3}}{\int }}f\left(x\right)dx+\underset{-\frac{1}{3}+\frac{\sqrt{10}}{3}}{\stackrel{2}{\int }}f\left(x\right)dx$$$$\mathrm{and}{\mathrm{that}}^{\prime }\mathrm{s}\mathrm{just}\mathrm{what}\mathrm{I}\mathrm{did}$$

Commented by Joel578 last updated on 03/Apr/18

$$\mathrm{You}\mathrm{are}\mathrm{right}.{\mathrm{It}}^{\prime }\mathrm{s}\mathrm{typo}.\mathrm{I}\mathrm{have}\mathrm{corrected}\mathrm{it}$$

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