Supplemental Instruction
Iowa State University / Leader: / Becca
Course: / Math 166
Instructor: / Dr. Martin
Date: / 10/2/12
Evaluate: (Questions 1-4 are part of the NON-CALCULATOR section)
Each question is worth 9 points.
1) 0πx2sinxdx
2) dr(r2+4)2
3) Find a function f(t) such that ftdt= t2t4-2t+8+C.
(READ THIS ONE CAREFULLY!)
4) Evaluate: 10z-40(z-3)(z2-4)dz
Questions 5-8 are part of the CALCULATOR section of the midterm:
5) For x ≥ 0, the graphs of y=fx and y=g(x) are above the x-axis and gx- fx=(x+1)e2x
Let R be the region between these two graphs for 1≤ x ≤6.
a. (6 points) Write an integral whose value is the area of R. DO NOT EVALUATE THE INTEGRAL.
b. (6 points) Write an integral whose value is equal to the volume of the solid generated when R is rotated about the line x = -3. DO NOT EVALUATE THE INTEGRAL.
c. (4 points) Suppose the region R is rotated about the x-axis to sweep out another solid. Is knowing gx- fx enough for you to calculate the volume of the solid? Explain.
6) (16 points) Professor Cosine and Professor Sine both evaluate the integral sin3t cos5t dt. Professor Cosine obtains:
sin3t cos5t dt= 18cos8t-16cos6t+C,
while Professor Sine’s result is:
sin3t cos5t dt= 18sin8t-13sin6t+ 14sin4t+C.
Are Professors Cosine and Sine both wrong? Are both answers correct? If not, who has a correct answer and who is incorrect? You must justify your answer with complete work and explanations.
7) (16 points) A storage facility is in the shape of a right circular cone of height 8 meters and base radius 4 meters. See the accompanying figure. The tank is filled with liquid that has a weight density of 600kg/m3. The top 2 meters of liquid in the tank is pumped to a level of 5 meters above the top of the tank. Calculate the work done.
8) (16 points) Let R be the region bounded by the graphs of y= x2 and y= x for 0≤x≤1. Find the coordinates of the centroid of R.