Hello everyone,

Could someone please check my work to see where I've erred? My answer is slightly below the one given by my textbook.

Thank you!

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39. Find the total areas of the shaded region in:

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My work:
I found the area separately, from left to right.

Shaded part on left = $\displaystyle \int_{-2}^{-1}(-x + 2 - 4 + x^2) dx $

$\displaystyle = \int_{-2}^{-1}(x^2 - x - 2) dx $ $\displaystyle = \frac{x^3}{3} - \frac{x^2}{2} - 2x\bigg|_{-2}^{-1} = 11/6 $

Shaded part in middle = $\displaystyle \int_{-{\color{red}1}}^{2}(4 - x^2{\color{red}+x-2}) dx - 7 $

$\displaystyle = 4x - \frac{x^3}{3} - 2x\bigg|_{2}^{-2} - 6 = \frac{14}{3} $

Shaded part on right = $\displaystyle \int_{2}^{3}(-x + 2 - 4 + x^2) dx {\color{blue}- \frac{1}{2}} $

$\displaystyle = (\frac{x^3}{3} - \frac{x^2}{2} - 2x\bigg|_{2}^{3}) {\color{blue}- \frac{1}{2}} = \frac{11}{6} {\color{blue}- \frac{1}{2}} = \frac{4}{3} $

Therefore, total area = [tex] \frac{47}{6} [\math]. According to textbook, total area =

49/6 (hidden in white).