Math 1B  Calculus II  Chapter 6 test (1)  fall ’06           Name__________________________

Show your work for credit.  Write all responses on separate paper.  Don’t abuse a calculator.

 

1.      Consider the area of the region bounded by  and .

a.       Sketch a graph illustrating the region.

b.      By integrating over x and using symmetry, as appropriate.

c.       By integrating over y and splitting the region into 2 pieces, as appropriate.

2.      Consider the area of the region bounded by  and .

a.       Sketch a graph illustrating the region.

b.      Compute the volume of revolution generated by revolving the region about the x-axis. 
Use either the shell method or the washer method, whichever seems easier.

c.       Compute the volume of revolution generated by revolving the region about the y-axis. 
Use either the shell method or the washer method, whichever seems easier.

3.      A hole of radius r is drilled through the center a sphere of radius R > r.  Find the volume of the remaining portion of the sphere.

4.      Consider the curve in the x-y plane described by the parametric equations

a.       Tabulate values of t, x, and y for t ranging from  to .

b.      Plot these points in the x-y plane and sketch the curve.

c.       Compute the area of the loop formed by this curve.  Integrate over t.

d.      Compute the arc length of the loop.  Integrate over t.

5.      Use Simpson’s rule with n = 4 to estimate the arc length of the parametric curve
                                              . 
for .  Note: there is no elementary antiderivative, so the approximation is appropriate.
Compare your answer with the value of the integral produced by your calculator.