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.
b. What is the volume of the solid that is generated when R is revolved about the y-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 \(x\) as a function of \(y\) from the hyperbola equation to describe the boundary curve. Solve for \(x\) to get \(x = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\), which represents the right branch of the hyperbola.
3. Set up the volume integral for the solid generated by revolving the region R about the y-axis. Since the region is bounded between \(x = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\) and \(x = c\), and \(y\) varies between appropriate limits, use the method of cylindrical shells or washers. Here, the washer method is convenient:
4. The volume element when revolving around the y-axis is given by \(dV = \pi (R_{outer}^{2} - R_{inner}^{2}) dy\), where \(R_{outer}\) and \(R_{inner}\) are the distances from the y-axis to the outer and inner boundaries of the region. In this case, \(R_{outer} = c\) and \(R_{inner} = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\). Determine the limits of integration for \(y\) by finding the intersection points of the hyperbola and the vertical line \(x = c\).
5. Write the volume integral as \(V = \pi \int_{y_{min}}^{y_{max}} \left(c^{2} - a^{2} \left(1 + \frac{y^{2}}{b^{2}}\right) \right) dy\). Evaluate this integral over the determined limits to find the volume of the solid.

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 for identifying the region R and setting integration limits.

Volume of Solids of Revolution

When a plane region is revolved around an axis, it generates a solid whose volume can be found using methods like the disk/washer or shell method. For revolution about the y-axis, the shell method is often convenient, integrating cylindrical shells formed by vertical slices of the region.

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 (x = c). Correctly identifying this vertical boundary is crucial to determine the limits of integration for the volume calculation, ensuring the volume corresponds exactly to the specified region.