Chapter 5: Integrals - Section 5.4 - The Fundamental Theorem of Calculus - Exercises 5.4 - Page 287: 51

$\pi$

The area of the rectangle with limits $0$ to $\pi$ is given as: $A=\int_0^{\pi} (1+\cos x ) dx$ $\implies [x+\sin x]_0^{\pi}=(\pi+\sin \pi)-(0+\sin 0)$ thus, $A=\pi$ Hence, the area of shaded region is: $ 2 \pi-\pi =\pi$