求阴影区域的面积。上图未按比例绘制。

1. Set up the definite integral for the area of the shaded region

$\text{Area} = \int_{-1}^{1} \left( 5e^{-2x} - (-6) \right) \, dx$
$\text{Area} = \int_{-1}^{1} \left( 5e^{-2x} + 6 \right) \, dx$

2. Integrate the expression with respect to $x$

$\int \left( 5e^{-2x} + 6 \right) \, dx = \left[ \frac{5e^{-2x}}{-2} + 6x \right]_{-1}^{1}$

3. Substitute the limits of integration

$= \left( -\frac{5}{2}e^{-2(1)} + 6(1) \right) - \left( -\frac{5}{2}e^{-2(-1)} + 6(-1) \right)$
$= \left( -\frac{5}{2}e^{-2} + 6 \right) - \left( -\frac{5}{2}e^{2} - 6 \right)$

4. Simplify to find the exact area

$= -\frac{5}{2}e^{-2} + 6 + \frac{5}{2}e^{2} + 6$
$= 12 + \frac{5}{2}e^2 - \frac{5}{2}e^{-2}$