1.(20)  Calculate the integrals.  Find the value if it exists or show that the integral diverges.  Show your work

a)  ò₁^¥[(x)/( 1+x²)]dx

b)  ò₁²[(ln(x-1))/( x-1)]dx

2.(20)  For -1 £ x £ 1, define   F(x) = ò_-2^x[Ö(4-t²)]dt

a)  What is the domain of F?  Explain why this is the domain.

b)  What does F(0) represent geometrically?

c)  What is the value of F(-2)?   F(0)?   F(2)?

d)  What is F ^¢(x)?

e)  What is [(d)/( dx)]ò_-2^x3[Ö(4-t²)]dt ?
 

3.(20)  Fact:  the force needed to stretch a spring is proportional to the amount the spring is stretched.

a) Give the general formula that the ''Fact'' states.  F = __________

Suppose a coiled spring is 1 foot long (with no stretching or compression).  It is then stretched to a total length of 1.25 feet (i.e., stretched ¹/₄ foot).  It takes 3 lbs of force to keep it stretched to this length.

b) Give the formula for the force F to stretch this spring x ft.:  F(x) = _______

c) What is the approximate amount of work done if the spring is stretched from a length of 1.5 ft to 1.501 ft?

d)  Calculate the amount of work done in stretching the spring from a length of 1.1 ft to a length of 2.4 ft.
 

4.(10)   a)  Set up an integral to find the arc length of the sine function over one period.

b)  Evaluate the integral by a method of your choice (substitution, Simpson, ¼) and tell what method you are using and why.
 

5.(20)  Consider the region D bounded on the top by the function y = 2+4x-x², on the bottom by y = 1, and on the sides by x = 1 and x = 3.

a)  Sketch the region D.  (Use your calculator.)

b)  Set up an integral that will give the volume of the region obtained by rotating D about the x-axis.  (Do not evaluate the integral.)

c)  Set up an integral that will give the volume of the region obtained by rotating D about the vertical line x = 5.  (Do not evaluate the integral.)
 

6.(10)  A baseball player signs a contract whereby he receives a sum that increases continuously and linearly from a starting salary of $1,000,000 a year and reaches $3,000,000 a year after 4 years.  Thus his salary after t years (in millions of dollars) is f(t) = 1+¹/₂t.  Set up an integral to find the present value of the contract assuming an interest rate of 6% per year compounded continuously.
 

Bonus: (5)  Evaluate the integral in #6 - by hand!