(a) Assume that a > 0. Evaluate dx, using the fact that the region bounded by y = x and the x-axis between 0 to a is a triangle. (See Figure 6.23.) Figure 6.23 The region for Problem 38. (b) Assume that a > 0. Evaluate by approximating the region bounded by y = x and the x-axis from 0 to a with rectangles. Use equal subintervals and take right endpoints. (Hint: Use the result in Example 3 to evaluate the sum of the areas of the rectangles.)

Using only integers between 0 and 10, construct two data sets with at least 10 observations each such that the two sets have the same mean, but different variances. Construct dot plots for each of your data sets, and mark the mean of each data set on its dot plot.

If $f$ is continuous and $\int_0^9 f(x)~dx = 4$, find $\int_0^3 xf(x^2)~dx$.

Consider a coal-fired steam power plant that produces $120\ \mathrm{MW}$ of electric power. The power plant operates on a simple ideal Rankine cycle with turbine inlet conditions of $9\ \mathrm{MPa}$ and $550^{\circ} \mathrm{C}$ and a condenser pressure of $15\ \mathrm{kPa}$. The coal has a heating value (energy released when the fuel is burned) of $29,300 \mathrm{~kJ} / \mathrm{kg}$. Assuming that $75$ percent of this energy is transferred to the steam in the boiler and that the electric generator has an efficiency of $96$ percent, find (a) the overall plant efficiency (the ratio of net electric power output to the energy input as fuel) and $(b)$ the required rate of coal supply.

Determine where f is continuous. Express the answer in interval notation. $f(x)=\sqrt[3]{4-x^{2}}$

The graph of the derivative $f'$ of a continuous function $f$ is given. On what intervals is $f$ concave upward? Concave downward?

Describe how you would construct each of the following. Refer to an obtuse $\triangle ABC$

The midpoint of $\overline{B C}$

Find the area of the rectangle with dimensions 0.5 m and 0.3 m.

A random sample of 31 eight-ounce servings of different juice drinks has a mean of 99.3 calories and a standard deviation of 41.5 calories. Use this information to construct the 90% and 95% confidence intervals for the population means. Which interval is wider? If convenient, use technology to construct the confidence intervals.

A typical adult human has approximately 5 L of blood in the body. If 1 ul (1 microliter) contains 5 x 10^6 red blood cell, how many red blood cells does a typical adult have? (Hint 1 L = 10^6 uL.)

The population density is $f(x, y)=1,000 y^{2} e^{-0.01 x}$ people per square mile at each point (x, y) within the region R bounded by the parabola $x = y^2$ and the vertical line x = 4. Find the total population in the region R.

Ticks are obligate blood feeders (meaning they must obtain blood to pass from one life stage to the next). Ticks attach to the outside of hosts where they suck blood and fluids and cause irritation. (a) Identify this type of interaction: _____________________________.

(b) How would the tick population be affected if the host became rare?

There are four basic human blood types, in order of frequency : O, A, B, and AB. a) Make a sample space showing the possible pairings of blood types for a mother and father. b) Why can't you use this sample space to find the prob- ability that a baby has parents whose blood types match?

Indicate the number of unpaired electrons present in each of the following atoms: B, Ne, P, Sc, Mn, Se, $\mathrm{Kr}$, Fe, $\mathrm{Cd}, \mathrm{I}$, and $\mathrm{Pb}$.