Math 1B  Fair Game for Chapter 6 Test

 

1.      The area of the plane bounded between  and  is revolved around the x-axis. Find the volume of revolution. 

a.       Sketch a graph of the region bounded and the volume of revolution.

b.      Set up an integral for the volume of revolution using the washer method.

c.       Set up an integral for the volume of revolution using the shell method.

d.      Evaluate one of these integrals.  Use a calculator if necessary.

 

2.      Find the length of the curve given by  and  for .  You will not need a calculator.

3.      The region bounded by  and  is revolved about the x-axis.  Set up an integral to compute the surface area of revolution.  Evaluate, using a calculator if necessary.

 

4.      Suppose the trough shown below is filled with water. The 10 meter length is vertical. And the front face is tilted at a  angle, as shown: 

a.       Find the minimum total work required to empty the water out through the top of the trough.

b.      Find the total fluid force on the slanted (front) face of the trough.

5.      Consider the region in the first quadrant bounded by  

a.       Find the x-coordinate of the centroid of this region.

b.      Suppose the volume generated by revolving this region about the y-axis is filled with water.  Set up an integral to compute the total fluid force on the surface of this volume.

6.      Suppose the lifetime (in days) of an electric light bulb is a random variable with a probability density function ,     

a.       Find the value of k that is needed for  to be a probability density function.

b.      Find the mean value for the lifetime of this electric light bulb.

c.       Express the probability that this light bulb lasts more than 100 days as an integral.

 

7.      Consider the region bounded by two circles with radii 7 and 8, where the circumference of the larger circle passes through the center of the smaller circle, as shown: a.       Introduce a coordinate system and sketch a graph showing how these circles are situated in your coordinate system. b.      Write equations for each circle relative to your coordinate system, either rectangular or parametric. c.       Set up an integral to compute the area of this region in terms of one of your coordinate variables or the coordinate parameter. Don't simplify the integral.

d.      Set up an integral to compute the volume generated by revolving this area around the line through the two circle centers using the 'shell' method.

e.       Set up an integral to compute the volume generated by revolving this area around the line through the two circle centers using the 'washer' method.

f.        Set up integrals for volume generated by revolving the area common to the circles around the line tangent to the larger circle and passing through the center of the smaller circle.  One integral using the washer method and the other using the shell method.  Use a CAS to check them.

 

8.      Consider the epitrochoid defined parametrically by
       
This is the curve traced out by a point on the circumference of a circle with radius 1 as it rolls around another circle of radius 3.

a.       Find the length of this curve.

b.      Set up integrals to find the centroid of the region bounded by the epitrochoid.

9.      Find the volume generated by revolving the area in the first quadrant bounded by  and  about the y-axis.

10.  Show that the length of the curve described by the equation  is given by the integral  where . 

11.  Suppose the volume generated by revolving the ellipse described by the parametric equations
                                               
around the y-axis is filled with water.  Simplify the integrand for an integral to compute the total fluid force on the surface of this volume.

12.  Two electrons r meters apart repel each other with a force of .  Suppose one electron is held fixed at the point (1,0) on the x-axis (units in meters).  How much work does it take to move a second electron along the x-axis form the point (-1,0) to the origin?

13.  A normal random variable X has the probability density function , where  is the mean and  is the standard deviation.  Simplify an integral for the probability  by making the substitution .  Approximate this integral using Simpson’s rule with n = 4.