Exercise:
Determina le aree delle figure evidenziate nei disegni presta attenzione al segno!vspac.cm center tikzpicture axisx.cmy.cm axis linesmiddle enlargelimitsfalse axis line styleshorten -pt shorten -pt xlabel styleanchorwest atticklabel* cs:. xshiftpt ylabel styleanchorsouth atticklabel* cs:. yshiftpt xlabel x ylabel y xmin- xmax ymin- ymax ytick--... xtick--... minor x tick num minor y tick num minor grid styleblack! major grid styleblack! grid yminorgrids xminorgrids major tick lengthpt every tick/.style semithick fillfillbluefill opacity.line width ptsmoothsamplesdomain: plotxx^-x^-*x -- -- -- cycle; fillfillbluefill opacity.line width ptsmoothsamplesdomain-: plotxx^-x^-*x -- -- - -- cycle; drawcolorblackline width ptsmoothsamplesdomain-: plotxx^-x^-*x; draw nodethickcolorblack inner sepptfillwhite yx^-x^-x; axis tikzpictureqquad tikzpicture axisx.cmy.cm axis linesmiddle enlargelimitsfalse axis line styleshorten -pt shorten -pt xlabel styleanchorwest atticklabel* cs:. xshiftpt ylabel styleanchorsouth atticklabel* cs:. yshiftpt xlabel x ylabel y xmin- xmax ymin- ymax ytick--... xtick--... minor x tick num minor y tick num minor grid styleblack! major grid styleblack! grid yminorgrids xminorgrids major tick lengthpt every tick/.style semithick fillfillbluefill opacity.line width ptsmoothsamplesdomain: plotx.*x^ -- -- -- cycle; fillfillbluefill opacity.line width ptsmoothsamplesdomain: plotx.*x^ -- . -- -- cycle; drawline width pt - -- ; drawcolorblackline width ptsmoothsamplesdomain-: plotx.*x^; drawcolorblackline width ptsmoothsamplesdomain-: plotx.*x+; draw .. nodethickcolorblack inner sepptfillwhite x; draw .. nodethickcolorblack inner sepptfillwhite yfracx+; draw . nodethickcolorblack inner sepptfillwhite yfracx^; axis tikzpicture center

Solution:
Bisogna stare attenti a non ergrare la funzione data sull'ero ervallo altrimenti le aree si sottraggono bensì spezzare l'egrale nei vari ervalli e poi sommare le aree ``positive''. Abbiamo che per il grafico di sinistra la primitiva è: x^-x^-xdxfracx^-fracx^-x^+C:Fx+C Dunque l'area è data da -F-F+F-F--F-F--left-fracright-left-fracrightfrac. Allo stesso modo per quella di destra sapo che il punto d'ersezione ha ascissa x. fracx^-fracx-dxfracx^-fracx^-x+C:Gx+C da cui l'area positiva è G-G-G-GG-G-frac- left-fracrightfrac.