A solid disk and a hoop are simultaneously released from rest at the top of an incline and roll down without slipping. Which object reaches the bottom first? The one that has the largest radius arrives first.

Solve $u_{x x}+u_{y y}=0$ in the disk $r<a$ with the boundary condition $\frac{\partial u}{\partial r}-h u=f(\theta)$ where $f(\theta)$ is an arbitrary function. Write the answer in terms of the Fourier coefficients of $f(\theta)$ .

Show that $4 \pi \leq \displaystyle \iint _ { D } \left( x ^ { 2 } + y ^ { 2 } + 1 \right) d x d y \leq 20 \pi$ where D is the disk of radius 2 centered at the origin.

A steel column of 5.5-m effective length must carry a centric dead load of 310 kN and a centric live load of 375 kN. Knowing that

$$σ_γ = 250 MPa$$

and E = 200 GPa, use load and resistance factor design to select the wide-flange shape of 310-mm nominal depth that should be used. The dead load factor

$$γ_D = 1.2,$$

the live load factor

$$γ_L = 1.6,$$

and the resistance factor Φ = 0.90.

Which type of microscope would be best to use to observe each of the following? a. a stained bacterial smear b. unstained bacterial cells: the cells are small, and no detail is needed c. unstained live tissue when it is desirable to see some intracellular detail d. a sample that emits light when illuminated with ultraviolet light e. intracellular detail of a cell that is 1 μm long f. unstained live cells in which intracellular structures are shown in color

Let D be the unit disk $x ^ { 2 } + y ^ { 2 } \leq 1$ . Evaluate $\displaystyle \iint _ { D } \exp \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$ by making a change of variables to polar coordinates.

Find the flux of $\vec{H} = 2\vec{i} + 3\vec{j} + 5\vec{k}$ through the surface $S$.

$S$ is the disk of radius $1$ in the plane $x + y + z = 1$ oriented in upward.

Let V be the volume of the solid bounded above by the hemisphere

$$z = \sqrt { 1 - r ^ { 2 } }$$

and bounded below by the disk enclosed within the circle r=sin

$$\theta$$

. Expressed as a double integral in polar coordinates, V=___.

A solid disk and a hoop are simultaneously released from rest at the top of an incline and roll down without slipping . Which object reaches the bottom first? The one that has the largest mass arrives first.

Evaluate the given double integral over the quarter-disk Q given by $x \geq 0, y \geq 0$, and $x^{2}+y^{2} \leq a^{2}$, where a > 0. $\iint_{Q} \frac{2 x y}{x^{2}+y^{2}} d A$

A thin circular disk of radius 12 cm has density which increases linearly from 1 gm/squared cm at the center to 25 gm/squared cm at the rim.

Evaluate the integral.

A compact disk (CD) speeds up uniformly from rest to $310 \mathrm{rpm}$ in $3.3 \mathrm{~s}$. Describe a strategy that allows you to calculate the number of revolutions the CD makes in this time.

Determine the couple M which must be applied at O in order to support the mechanism in the position $\theta=30^{\circ}$. The masses of the disk at C, bar OA, and bar BC are $m_{0}$, m, and 2m, respectively.

Would you use a sample or a census to measure each of the following? Why? If you are uncertain, explain the issues.

The percentage of disk space available on laptop computers owned by your five closest friends.