   Chapter 10.5, Problem 61E

Chapter
Section
Textbook Problem

Find the area of the region enclosed by the hyperbola x2/a2 − y2/b2 = 1 and the vertical line through a focus.

To determine

To find: The area of the region enclosed by the hyperbola x2a2y2b2=1 and the vertical line through a focus.

Explanation

Given:

The area of the region enclosed by the hyperbola and the vertical line through a focus is shown in figure(1) below.

The equation of the hyperbola is x2a2y2b2=1

Calculation:

Let, rewrite the equation of hyperbola as below,

y2b2=(1x2a2)y2b2=(1x2a2)y2b2=(1+x2a2)

y2b2=(a2+x2a2)y2=(x2a2a2)×b2y2=b2a2(x2a2)

y=b2a2(x2a2)y=±bax2a2

Compute the area of the region enclosed by hyperbola.

A=2acydx

Substitute bax2a2 for y .

A=2ac(bax2a2)dx=ba2(c2c2aa2ln|c2+c2c2a2ac|)a2=ba2(c2c2a2a2ln|c2+c2c2a2ac|)a2

=ba(cc2a2a2ln|c2+cc2a2ac|)a2=ba(cc2a2<

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