Exercises

01

Surface area of a torus

A circle in the $xz$-plane, offset on the $x$-axis by $+R$ from the origin, satisfies the equation $(x-R{)}^2+z^2=r^2$. When this circle is rotated about the $z$-axis, the result is a torus. To find the equation it satisfies, replace the variable $x$ with the radial quantity $\sqrt{x^2+y^2}$ in the $xy$-plane:

$${\left(\sqrt{x^2+y^2}-R\right)}^2+z^2=r^2.$$

This torus may be described parametrically by the formula:

$$\mathbf{r}(u,v)=((R+r\cos (u))\cos (v),(R+r\cos (u))\sin (v),r\sin (u)).$$

Calculate the surface area of this torus. Your answer should be a simple expression in terms of $R$ and $r$.

02

Theorem of Archimedes: band of a sphere

A band of a sphere is formed by taking the region of the sphere lying between two horizontal planes. Archimedes discovered that the area of a band of a sphere is equal to the area of the corresponding band of a cylinder that circumscribes the sphere. Prove this result. Use a sphere centered at the origin of radius $R$, a vertical cylinder at the origin of radius $R$, and horizontal planes given by $z=a$ and $z=b$.

(Hint: you will need to solve for the range of azimuthal angles $\phi \in [{\phi }_0,{\phi }_1]$ that corresponds to the range $z\in [a,b]$ by using the last component function of the parametrization of the sphere.)

03

New variables to improve the bounds

Let $\mathcal{D}$ be the domain given by $xy\in [1,4]$ and $y/x\in [1,4]$. Change variables and then compute the integral:

$${\iint }_{\mathcal{D}}{x}^2+y^{2}dxdy.$$

(Hint: let $u^2=xy$ and $v^2=y/x$, so $x=u/v$ and $y=uv$.)