MATH115 GEOMETRY AND CALCULUS

Bonus Exercises 5

No homework! These problems are just for practice!

• True or false?

  • The integral ${\int }_1^{\mathrm{\infty }}f(x)𝑑x$ converges if and only if $f(x)\to 0$ as $x\to \mathrm{\infty }$.

  • If a vector-valued function $𝐟(x,y,z)=(f_1,f_2,f_3)$ can be expressed as $\nabla \varphi$ for some $\varphi (x,y,z)$, then so can $({(f_1)}_x,{(f_2)}_y,{(f_3)}_z)$.

• A tetrahedron $B$ has vertices $(0,0,0)$, $(6,0,0)$, $(0,8,0)$ and $(0,0,24).$

  • Check that $(6,0,0)$, $(0,8,0)$ and $(0,0,24)$ lie on the plane that has equation

    $$z=24-4x-3y.$$

  • Show that the intersection of $B$ with the plane $\{(x,y,z):z=0\}$ is given by the triangle

    $$R=\{(x,y):0\le x\le 6;0\le y\le 8-\frac{4x}{3}\}.$$

  • Calculate the volume of $B$ by finding $\int {\int }_{R}z𝑑y𝑑x$.

• Let $y=f(x)=\sqrt{r^2-x^2}$, $a\le x\le a+h$, be rotated about the $x$-axis. The surface obtained is a region of the sphere of radius $r$ lying between two vertical and parallel planes. Find its surface area (using $y^{\prime }=\frac{-x}{\sqrt{r^2-x^2}}$).