Question

1. let $mathcal{u}$ be the solid contained in $x^{2}+y^{2}-z^{2}=16$ and lying between the planes $z = - 3$ and $z = 3$. sketch $mathcal{u}$ and write an iterated integral which expresses its volume. in which orders of integration can you write just a single iterated integral (as opposed to a sum of iterated integrals)?

Explanation:

Step1: Rewrite the surface equation

The surface is $x^{2}+y^{2}=z^{2} + 16$, which is a hyperboloid of one - sheet. For a fixed $z$ value between $z=-3$ and $z = 3$, the cross - section in the $xy$ - plane is a circle with equation $x^{2}+y^{2}=z^{2}+16$. The radius of the circle at height $z$ is $r(z)=\sqrt{z^{2}+16}$.

Step2: Set up the volume integral in $dzdA$ order

The volume of a solid $V=\iiint_{U}dV$. Using the order $dzdA$ (where $dA = dxdy$), and since $z$ ranges from $z=-3$ to $z = 3$, and for each $z$ the region in the $xy$ - plane is a circle $x^{2}+y^{2}\leq z^{2}+16$. We can convert to polar coordinates. In polar coordinates, $x = r\cos\theta$, $y=r\sin\theta$, and $dA=rdrd\theta$. The limits for $r$ are from $0$ to $\sqrt{z^{2}+16}$ and for $\theta$ are from $0$ to $2\pi$. The volume integral is $V=\int_{-3}^{3}\int_{0}^{2\pi}\int_{0}^{\sqrt{z^{2}+16}}rdrd\theta dz$.

Step3: Analyze single - iterated integral orders

We can also use the order $dxdydz$. The limits for $z$ are from $z=-3$ to $z = 3$. For a fixed $z$, the region in the $xy$ - plane is given by $x^{2}+y^{2}\leq z^{2}+16$, so $y$ ranges from $y =-\sqrt{z^{2}+16 - x^{2}}$ to $y=\sqrt{z^{2}+16 - x^{2}}$ and $x$ ranges from $x=-\sqrt{z^{2}+16}$ to $x=\sqrt{z^{2}+16}$. The volume integral is $V=\int_{-3}^{3}\int_{-\sqrt{z^{2}+16}}^{\sqrt{z^{2}+16}}\int_{-\sqrt{z^{2}+16 - x^{2}}}^{\sqrt{z^{2}+16 - x^{2}}}dydxdz$. We can write a single iterated integral in the orders $dxdydz$ and $dzdA$ (where $dA$ is in polar coordinates).

Answer:

The volume integral in $dzdA$ (polar coordinates) is $V=\int_{-3}^{3}\int_{0}^{2\pi}\int_{0}^{\sqrt{z^{2}+16}}rdrd\theta dz$. We can write a single iterated integral in the orders $dxdydz$ and $dzdA$ (polar coordinates).