The region between the curve $y=1 /(2 \sqrt{x})$ and the $x$-axis from $x=1 / 4$ to $x=4$ is revolved about the $x$-axis to generate a solid. Find the volume of the solid.

A$3-\mathrm{ft}$-long steel pipe is subjected to a torque of $1200\ \mathrm{lb} \cdot \mathrm{ft}$ at each end. The pipe is made of $0.2 \% \mathrm{C}$ hardened steel, failure is by yielding, and the factor of safety is $1.5$. Find the nominal diameter of the lightest standard-weight steel pipe that can be used for the shaft.

How many straight lines that pass through the origin are tangent to y = cos x? Find (to 6 decimal places) the slopes of the two such lines that have the largest positive slopes.

What is the total number of electrons in the sixth energy level (n = 6)?

Vocabulary: Sex ratio

Explaining the clues you use, make inferences about the speaker's character in (a) lines $31$-$43$ of "My Last Duchess" and (b) lines $18$- $20$ of "Life in a Love."

Without graphing, decide.

a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point?

b. How many solutions does the system have? See the discussed examples.

$$\left\{\begin{array}{l}x+y=4 \\ x+y=3\end{array}\right.$$

A peacock mantis shrimp smashes its prey with a hammer-like appendage by storing energy in a springlike section of exoskeleton on its appendage. The spring has a force constant of $5.9 \times 10^4 \mathrm{~N} / \mathrm{m}$. What power does the shrimp generate if it unleashes the energy in $0.0018$ s? (The power per mass of the shrimp appendage is more than 20 times greater than that generated by muscle alone, showcasing the shrimp's ability to harness the potential energy of its spring.)

Choose the letter of the best answer. What determines a single inherited trait of an individual? A. one pair or many pairs of genes B. a disrupted cell cycle C. one half of the replicated DNA strand D. amino acids joined to form proteins

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ be a random sample from a normal distribution. Use the statement to prove that $S^{2}$ is consistent for $\sigma^{2}$.

_____ open water area of a lake or pond that is well lit and dominated by plankton.

A torque of $650\ \mathrm{Nm}$ rotates a shaft of diameter $0.25 \mathrm{~m}$ with $\omega=50\ \mathrm{rad} / \mathrm{s}$. What are the shaft surface speed and the transmitted power?

Integrate f over the given region.

$$f ( x , y ) = x ^ { 2 } + y ^ { 2 }$$

over the triangular region with vertices (0, 0), (1, 0), and (0, 1)

How did Thomson determine that the electrons have a negative charge?