Chapter 14 - Multiple Integrals - 14.2 Double Integrals Over Nonrectangular Regions - Exercises Set 14.2 - Page 1016: 32

$\sinh 1-\cosh 1+1$

Using the rules of integrals, we obtain: Area $=\iint_{R} d A=\int_{0}^{1} \int_{\sinh x}^{\cosh x} d y d x$ $=\int_{0}^{1}[y]_{\sinh x}^{\cosh x} d x$ $=\int_{0}^{1} -\sinh x +\cosh x d x$ $=[-\cosh x+\sinh x]_{0}^{1}$ $=[\sinh 1-\cosh 1]-[\sinh 0-\cosh 0]$ $=\sinh 1-\cosh 1+1$