Find the surface area generated by rotating the lemniscate about the line .

Question

Find the surface area generated by rotating the lemniscate  about the line  .

Quizplus · Accepted Answer

The lemniscate is given by the equation $(x^2 + y^2)^2 = 2(3^2 - x^2 - y^2)$ which can be simplified to $x^4 + y^4 + 2x^2y^2 = 18 - 2x^2 - 2y^2$.

The line of rotation is given by $y = -x$.

Using the formula for the surface area of a solid of revolution, we have:

$$\begin{aligned} S &= 2\pi\int_{-3}^{3} x\sqrt{1 + (-1)^2}\sqrt{x^4 + y^4 + 2x^2y^2 - 18}\ &= 2\pi\int_{-3}^{3} x\sqrt{x^4 + 2x^2(-x^2) - 17}\ dx\ &= 2\pi\int_{-3}^{3} x\sqrt{3x^4 - 17}\ dx. \end{aligned}$$

This integral can be evaluated by substituting $u = 3x^4 - 17$ and integrating by parts. After some algebra, we find that the surface area is:

$$S = \frac{32}{15}\pi(17^{3/2} - 1).$$

Simplifying this expression gives us the answer B.

Find the Surface Area Generated by Rotating the Lemniscate