Question

Calcule a área da região a seguir

Answer 1

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A área total é a soma das regiões I e II.

$\tt\acute{A}rea~da~regi\tilde{a}o~I:\\\displaystyle\sf \int_{0}^{1}e^x~dx=e^x\Bigg|_{0}^{1}=e^{1}-e^{0}=e-1\checkmark$

$\tt\acute{A}rea~da~regi\tilde{a}o~II:\\\displaystyle\sf\int_{1}^{e}(e-\ell n(x))~dx=ex-\left[x\cdot\ell n(x)-x\right]\Bigg|_{1}^{e}\\\sf =e\cdot e-\left[e\cdot\ell n(e)-e\right]-\left[e\cdot1-\left[1\cdot\ell n(1)-1\right]\right]\\\sf=e^2-e\ell n(e)+e-e+\ell n(1)-1\\\sf=e^2-\diagup\!\!\!e+\diagup\!\!\!e-e+0-1\\\displaystyle\sf\int_{1}^{e}(e-\ell n(x))~dx=e^2-e-1$

$\sf\acute{a}rea~total=e-1+(e^2-e-1)\\\sf\acute{a}rea~total=\diagup\!\!\!e-1+e^2-\diagup\!\!\!e-1=e^2-2\checkmark$