Problem 98AE

The anvil of a hyperbola Let H be the hyperbola x2 − y2 = 1 and let S be the 2-by-2 square bisected by the asymptotes of H. Let R be the anvil-shaped region bounded by the hyperbola and the horizontal lines y = ±p (sec figure).

a. For what value of p is the area of R equal to the area of S?

b. For what value of p is the area of R twice the area of S?

Solution 98AE

Step 1:

We find the area of region R by integrating the hyperbola with respect to y from .

Equation of hyperbola in terms of y is :

Now, area is