Question

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?

Accepted Answer

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 $$x = c. $$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. 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: 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 $$x = c. $$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.

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 video answer for a similar problem:

Key Concepts

Equation and Properties of a Hyperbola

Volume of Solids of Revolution

Setting Integration Limits Using the Focus and Boundary Lines

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 video answer for a similar problem:

Key Concepts

Equation and Properties of a Hyperbola

Volume of Solids of Revolution

Setting Integration Limits Using the Focus and Boundary Lines

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?