MATH 202 PRACTICE MIDTERM 2

 

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

Printable Key

PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true explain why. 

A)   If all six limits of integration of an integral written in spherical coordinates are constants, then the region of integration is a sphere.

Solution

B)   

  Solution

PROBLEM 2 Set up integrals to evaluate the following.  Use the coordinate system that will most effectively solve the integral. 

A.     The mass of the tetrahedron that lies in the first octant and below the plane x + y + 2z  =  2  such that the density function is f(x,y,z)  =  3yz .

  Solution

B.     The surface area of the part of the paraboloid  z  =  9 - x2 - y2   that lies above the plane z = 5 .

  Solution

C.     The moment of inertia about the z-axis of the solid between the cylinders  x2 + y2  =  25  and  x2 + z2  =  25 that has density function 1.

  Solution

PROBLEM 3

Switch the order of integration. 

       

Solution

PROBLEM 4

A master dart thrower has determined that her probability density function is inversely proportional to one more than the fourth power of the distance in centimeters from the center of the dartboard. 

  1. Find the constant of proportionality.  (Hint:    )  

Solution

  1. Find the probability of her hitting the bullís-eye which is one centimeter in radius.

  Solution

PROBLEM 5

Consider the solid described by

        (x + 2y)2 + (2y - 2z)2 + (x + z)2  < 25    

with density function 

        f(x,y)  =  x + 2y                 

Show that the mass of the solid is equal to 

                       

Hint:  Recall that  

Solution