Chapter 6.7, Problem 70E

Show that the area of the shaded hyperbolic sector in Figure 7 is $A\left(t\right)=\frac{1}{2}t$. [Hint: First show that

$A\left(t\right)=\frac{1}{2}\sinh t\cosh t-{\displaystyle {\int }_1^{\cosh t}\sqrt{x^2-1}dx}$

and then verify that $A^{\prime }\left(t\right)=\frac{1}{2}$.]

FIGURE 7