Mth252 Test#3 Review Ch5.8,5.9,5.10,6.1,6.2,6.3,6.4,6.5
Ch5.8 Integration Using Tables and Computer Algebra Systems
During the test you may use your calculator, and Maple.
Use tables to find the antiderivative. Compare to using TI89 or Maple.
1a. 1b.
Ch5.9 Approximate Integration
2. To estimate the surface area of a pond, a surveyor takes several measurements, as show in figure 1.
Estimate the surface area of this pond using Simpson’s Rule.
figure 1.
Ch5.10 Improper Integrals
3. Determine whether each integral is convergent or divergent using a test. Evaluate those that are convergent
a) b) c)
4. Use the comparison test to determine whether is convergent or divergent.
Ch6.1 More About Areas
5a Find the area of the shaded region in figure 2
figure 2
5b The curve traced out by a point at a distance 1 m from the center of a circle of radius 2m as the circle rolls along the x-axis is called a trochoid and has parametric equations
.
One arch is given by . Find the area under one arch of this trochoid.
Ch6.2 Volumes
6. Find the volume of the solid obtained when the region bounded by the lines
and the x-axis is rotated about the y-axis.
7. A tank on the wing of a jet is formed by revolving the region bounded by the graph of and the x axis about the x-axis, where x and y are measured in meters. Find the volume of the tank.
8. Find the volume of the solid whose base is the region bounded by the lines , , and x=0. The cross sections perpendiculars to the x-axis are equilateral triangles.
9. Find the volume of the solid formed by the region in the first quadrant bounded by the graph of
and they-axis revolved about the x-axis.
Ch6.3 Arc Length
10. Find the length of the curve Use your calculator to make a sketch of the curve.
Ch6.4 Average Value of a Function
11. Suppose a study suggests that between the hours of 1:00 p.m. and 4:00 p.m. on a normal weekday, the speed of the traffic on US26 at 185th exit is modeled by km/hr where t is hours past noon. Compute the average speed of the traffic between 1 p.m. and 4 p.m. Sketch the graph. Illustrate the solution geometrically.
Ch6.5 Applications to Physics and Engineering
12. A force of 750 pounds compresses a spring 3 inches from its natural length of 15 inches. Find the work done in compressing the spring an additional 3 inches.
13. A 20-foot chain, weighing 5 pounds per foot, is lying coiled on the ground. How much work is required to raise one end of the chain to a height o 20 feet so that it if fully extended.
14. A quantity of gas with an initial volume of 1 cubic foot and pressure of 500 pounds per square foot expands to a volume of 2 cubic feet. Find the work done by the gas. Assume the pressure is inversely proportional to the volume. Use the formula where P is the pressure and v is the volume.
15. A tank in the shape of a half hemisphere is full of water. The radius of the tank is 5 feet. Water weighs 62.5 lb/ft3. Find the work required to pump the water out of the tank.
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