91Ó°ÊÓ

One hundred bacteria are started in a culture and the number \(N\) of bacteria is counted each hour for 5 hours. The results are shown in the table, where \(t\) is the time in hours. $$ \begin{array}{|l|c|c|c|c|c|c|}\hline t & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline N & 100 & 126 & 151 & 198 & 243 & 297 \\\\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find an exponential model for the data. (b) Use the model to estimate the time required for the population to quadruple in size.

Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=x^{2}, y=x $$

The region bounded by \(y=\sqrt{x}, y=0, x=0,\) and \(x=4\) is revolved about the \(x\) -axis. (a) Find the value of \(x\) in the interval [0,4] that divides the solid into two parts of equal volume. (b) Find the values of \(x\) in the interval [0,4] that divide the solid into three parts of equal volume.

A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

Bulb Design An ornamental light bulb is designed by revolving the graph of \(y=\frac{1}{3} x^{1 / 2}-x^{3 / 2}, \quad 0 \leq x \leq \frac{1}{3}\) about the \(x\) -axis, where \(x\) and \(y\) are measured in feet (see figure). Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Assume that the glass is 0.015 inch thick.)

Volume of a Storage Shed A storage shed has a circular base of diameter 80 feet (see figure). Starting at the center, the interior height is measured every 10 feet and recorded in the table. \begin{tabular}{|l|c|c|c|c|c|} \hline\(x\) & 0 & 10 & 20 & 30 & 40 \\ \hline Height & 50 & 45 & 40 & 20 & 0 \\ \hline \end{tabular} (a) Use Simpson's Rule to approximate the volume of the shed. (b) Note that the roof line consists of two line segments. Find the equations of the line segments and use integration to find the volume of the shed.

In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \((x-5)^{2}+y^{2}=16\) about the \(y\) -axis

The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\). Explain why this is so. Use symmetry to write a single integral that does represent the area.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graphs of \(f\) and \(g\) intersect midway between \(x=a\) and \(x=b,\) then \(\int_{a}^{b}[f(x)-g(x)] d x=0\)