- #1

songoku

- 2,340

- 340

- Homework Statement
- Please see below

- Relevant Equations
- Double Integral

Triple Integral

Cylindrical coordinate

Spherical coordinate

My attempt:

The shape of the hyperboloid would be like this:

If the hyperbolod is cut by plane z = d, the intersection would be a ellipse. Projecting the intersection to xy - plane, I think I get:

$$-2\leq x \leq 2$$

$$-b\sqrt{1-\frac{x^2}{a^2}} \leq y \leq b\sqrt{1-\frac{x^2}{a^2}}$$

So the volume would be:

$$V=2 \times \int_{-2}^{2} \int_{-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}} \left(c\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}-1}\right)dydx$$

Is this correct?

Thanks

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