find the exact area of the shaded region.

Explanation:

Step1: Recall area - between - curves formula

The area $A$ between two curves $y = f_1(x)$ and $y = f_2(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}[f_1(x)-f_2(x)]dx$. Here, $f_1(x)=\sin(x)+1$, $f_2(x)= - 2$, $a = 0$, and $b=\pi$.

Step2: Set up the integral

$A=\int_{0}^{\pi}[(\sin(x)+1)-(-2)]dx=\int_{0}^{\pi}(\sin(x)+1 + 2)dx=\int_{0}^{\pi}(\sin(x)+3)dx$.

Step3: Integrate term - by - term

We know that $\int\sin(x)dx=-\cos(x)+C$ and $\int 3dx = 3x+C$. So, $\int_{0}^{\pi}(\sin(x)+3)dx=\left[-\cos(x)+3x\right]_{0}^{\pi}$.

Step4: Evaluate the definite integral

First, substitute $x = \pi$: $-\cos(\pi)+3\pi=-(-1)+3\pi=1 + 3\pi$. Then substitute $x = 0$: $-\cos(0)+3\times0=-1+0=-1$. Then subtract: $(1 + 3\pi)-(-1)=2 + 3\pi$.

Answer:

$2 + 3\pi$