Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus
a. What is the area of R?

Verified step by step guidance

1. Identify the given hyperbola equation: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The right branch corresponds to \(x \geq a\) since the hyperbola opens along the x-axis.
2. Recall that the foci of the hyperbola are located at \(x = \pm c\), where \(c = \sqrt{a^2 + b^2}\). The right focus is at \(x = c\).
3. Set up the integral for the area \(R\) bounded by the hyperbola and the vertical line through the right focus \(x = c\). The area can be expressed as an integral with respect to \(x\) from \(x = a\) to \(x = c\):
4. \[\text{Area} = \int_a^c y(x) \, dx,\] where \(y(x)\) is the positive branch of the hyperbola.
5. Solve for \(y\) from the hyperbola equation: \(y = \frac{b}{a} \sqrt{x^2 - a^2}\). Substitute this into the integral to get:
6. \[\text{Area} = \int_a^c \frac{b}{a} \sqrt{x^2 - a^2} \, dx.\]
7. Evaluate the integral using an appropriate substitution, such as \(x = a \sec \theta\), to simplify the square root and compute the definite integral from \(x = a\) to \(x = c\).

Key Concepts

Equation and Geometry of a Hyperbola

A hyperbola is defined by the equation x²/a² - y²/b² = 1, representing two symmetric branches. The right branch corresponds to x ≥ a. Understanding the shape and position of the hyperbola, including its vertices and foci, is essential to identify the region bounded by the curve and the vertical line through the focus.

Coordinates of the Focus of a Hyperbola

The foci of the hyperbola x²/a² - y²/b² = 1 lie at (±c, 0), where c = √(a² + b²). The vertical line through the right focus is x = c. Knowing the focus location helps define the boundary of the region R and sets the limits for integration when calculating the area.

Area Calculation Using Definite Integrals

The area of region R can be found by integrating the function defining the hyperbola's upper branch between x = a and x = c. This involves setting up the integral of y = (b/a)√(x² - a²) with respect to x, then evaluating it to find the exact area bounded by the curve and the vertical line.