Expert: Sherry Wallin - 2/19/2009

Question
QUESTION: Find the average value of f(x) = cos²x on the interval [0, pi/2]

ANSWER: Hi Tyson~
    One definition of the average value of a function is:
1/(b-a)int(b to a)f(x) dx, so this translates into:
[1/(pi/2)]int(0, pi/2) cos^2x dx. Using a table of integrals you will find that the integral of cos^2x is (1/2)x+(1/4)sin2x so...
[1/(pi/2)][(1/2)x+(1/4)sin2x] evaluated from 0 to pi/2
= [1/(pi/2)][(pi/4)+(1/4)sin pi - [(1/2)*0+(1/4)sin 0]]
= [1/(pi/2)](pi/4)+ 0-0-0= (pi/4)/(pi/2) = pi/2

so there is your answer Tyson.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: A 144lb bag of sand is lifted at a constant rate. At 18ft half the sand is gone. How much work has been done?

Answer
This sounds like a Physics problem. Work can be defined as the dot product of the force and the displacement vectors so
W = F*D = fd cos(phi)where I've used the asterisk to mean 'dot' and the capital letters to represent vectors, and phi is the angle between the force and the displacement vectors. The displacement is the distance times the cos of the angle. Here in this problem, I am assuming the angle is zero, i.e., the displacement and the force are in the same direction. In order to lift the bag the force must over come gravity and the mass of the bag of sand. The gravitational force is -32lb/sec^2 and the weight of the bag is 144lbs. But you know that F = ma and here W = mg. So the amount of work is just 144lbs*ft/sec^2(18ft) = 1152ft/sec^2.

Hopefully I have explained the route I have taken and why I chose it to give you a better understanding.

Math Prof