Volume of a hyperbolic cap 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 volume of the solid that is generated when R is revolved about the x-axis?

Verified step by step guidance

1. Identify the region R bounded by the right branch of the hyperbola given by the equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) and the vertical line through the right focus. Recall that the foci of the hyperbola are located at \(x = \pm c\), where \(c = \sqrt{a^{2} + b^{2}}\). So the vertical line is \(x = c\).
2. Express \(y\) as a function of \(x\) from the hyperbola equation. Solve for \(y\) to get \(y = \pm \frac{b}{a} \sqrt{x^{2} - a^{2}}\). Since the region is bounded by the right branch, consider \(x \geq a\) and the positive \(y\) values for the upper half (or use symmetry if needed).
3. Set up the volume integral using the method of disks or washers for revolution about the x-axis. The volume element is \(dV = \pi y^{2} dx\), so the volume \(V\) is given by the integral \(V = \pi \int_{a}^{c} y^{2} \, dx\).
4. Substitute \(y^{2}\) from the expression found: \(y^{2} = \left( \frac{b}{a} \right)^{2} (x^{2} - a^{2})\). Thus, the integral becomes \(V = \pi \int_{a}^{c} \left( \frac{b^{2}}{a^{2}} (x^{2} - a^{2}) \right) dx\).
5. Evaluate the integral by integrating term-by-term: \(\int_{a}^{c} (x^{2} - a^{2}) dx = \int_{a}^{c} x^{2} dx - a^{2} \int_{a}^{c} dx\). After integration, multiply by \(\pi \frac{b^{2}}{a^{2}}\) to find the volume expression.

Key Concepts

Equation and Properties of a Hyperbola

A hyperbola is defined by the equation x²/a² - y²/b² = 1, representing two branches. The right branch corresponds to x ≥ a. Understanding the shape and position of the hyperbola, including its foci located at (±c, 0) where c² = a² + b², is essential to identify the region R and set integration limits.

Volume of Solids of Revolution

When a plane region is revolved around an axis, the volume of the resulting solid can be found using methods like the disk/washer or shell method. For revolution about the x-axis, the disk method integrates π[y(x)]² dx over the interval, where y(x) is the function describing the boundary curve.

Setting Integration Limits Using the Focus and Boundary Lines

The region R is bounded by the hyperbola and the vertical line through the right focus at x = c. Correctly identifying this vertical boundary determines the integration limits for the volume calculation. This step ensures the volume corresponds exactly to the specified hyperbolic cap.