Exam 6: Applications of the Definite Integral

The annual increase in the value of a particular investment is , where A₀ is the amount of money initially invested, i is the interest rate expressed as a fraction or decimal (not a percent), and t is the time over which the investment has been growing. While the value of this investment increases over time as indicated, the amount of goods and services that money could buy does not increase as rapidly because of inflation. Inflation reduces the buying power of the investment annually by , where d is the rate of inflation expressed as a fraction. If i = 0.07/yr and d = 0.01/yr, how much additional buying power will an initial investment of $10,000 earn over ten years? Round to the nearest dollar.

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

What is the total hydrostatic force on the walls of a cylindrical water tower 40' high and having a radius of 21'? [The density of water is 62.4 lbs/ft³.]

A given sample of a particular polymer will have molecules with a range of molecular weights, MW. If the molecular weight pdf is , what fraction of the molecules will have a molecular weight greater than 10,000?

Let R be the region bounded by . Compute the volume of the solid formed by revolving R about y = 15.

Identify the center of mass for a thin wire with density .

Find the volume of the solid formed by revolving the region bounded by about x = 7.

Given a certain amount of land potentially devoted to farming, one will first use the more fertile regions where production costs are lowest before using the less fertile regions. The consequence of this logical strategy is that the production costs increase as production increases. A certain landowner has enough land to produce 10⁶ bushels of a particular crop, and the production costs, c, of the n^th bushel is described with the equation . If the market price is p for this particular crop, the profit for producing the n^th bushel is . If p = $5.50, how much total profit could the farmer expect to obtain by producing bushels.

Find the volume of the solid with cross-sectional area extending over the range .

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 15% of the dirt?

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 20 feet? [The density of water is 62.4 lbs/ft³.]

Where is the center of mass of a region of uniform density bounded by ?

Sketch the solid bounded by and the x-axis on the interval revolved about the line Draw a typical shell and write an integral that can be used to compute the volume of the solid.

Find the area between the following curves on the given interval.

An object is dropped from a height of 100 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 50 ft/s. Determine when and how high up the objects collide.

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Starting with the expression for work , one can change the look of the expression without changing what it represents by dividing the force F by the area it is applied over and multiplying the differential distance dx by that same area. Force divided by area is a pressure (P), and area times the differential distance is a differential volume. Hence one can also describe work as . A gas at constant temperature will change its pressure inversely to changes in volume, . If a sample of gas has a pressure of 1 atmosphere when its volume is 1 L, how much work does it do when it expands from 1 L to 4 L?

A given sample of a particular polymer will have molecules with a range of molecular weights, MW. If the molecular weight pdf is , what fraction of the molecules will have a molecular weight between 50 and 100?

The base of a solid V is the region bounded by and Find the volume if V has square cross sections.

Is the function a probability density function on the interval ?