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.
a. What is the volume of the solid that is generated when R is revolved about the x-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 $$y as a $$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). 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}$$ \, dx. $$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}$$) \right) dx$$. 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.

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?

Risposta video verificata per un problema simile:

Concetti chiave

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.a. What is the volume of the solid that is generated when R is revolved about the x-axis?

Risposta video verificata per un problema simile:

Concetti chiave

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.
a. What is the volume of the solid that is generated when R is revolved about the x-axis?