Expert: Abe Mantell - 9/3/2005

Question
Hello. I would appreciate it if you could check my answers. I'm sorry it is so long.

1.) Let R denote the region between the curves y=x^-1 and y=x^-2 over the interval 1<= x <= 10.

a. Set up an integral for the area of R.
My answer: 1.403

b. Find x-bar, the x coordinate of the centroid of R.
My answer: 4.775

c. Set up and evaluate an integral for the volume of revolution of the solid generated when R is revolved about

i. the x-axis
My answer: 1.781

ii. the y-axis
My answer: infinity

2.) The length of a cable is 50 and the weight is 10. A portion of length 40 was hanging over the edge of a tall building and was pulled to the top. How much work was done?
My answer: 3920

3.) Let C denote the curve y= x(4-x), where 0<= x <= 4. Set up the integral for the following. In this case, do not evaluate the integrals.

a. the length of C
My answer: integral from 0 to 4 of sqrt[ 1+ (4-2x)^2] dx

b. the area of the surface generated when C is revolved about

i. the x-axis
My answer: 2pi *integral from 0 to 4 of x(4-x)*sqrt[1+ (4-2x)^2] dx

ii. the y-axis
My answer: 2pi* integral from 0 to 4 of [sqrt(4-y) -2] *sqrt[1+ (-2*sqrt(4-y))^-2] dy

4.) A tank has the shape of a trapezoidal prism. The top is horizontal and the two ends are vertical. The length is 4. The height is 2. The top is a 3-by-4 rectangle. Viewed from an end, the tank looks like the trapezoid shown in the figure below. Assume the tank contains a liquid to a depth of 1. Take the density of the liquid to be p.

a. Set up, but do not evaluate, an integral for the work required to pump the liquid to the top of the tank.
My answer: W= p*integral from 0 to 2 of 4 dy

b. Set up, but for not evaluate, an integral for the fluid force against one end of the tank.
My answer: F= integral from 0 to 4 of p(24) dx


http://img361.imageshack.us/img361/559/calctest3nq.png

My work:

http://img382.imageshack.us/img382/2565/calcscan5ae.jpg

Thanks


Answer
Hello Shay,

You have quite a few questions!  Let me just help you
with the first two for now...if you still need help
with the others, then you can resend them.

1 a) correct  b) correct
- c) Both are incorrect!
- (i) is NG because you squared (1/x+1/x^2) incorrectly
- you didn't "FOIL"
- (ii) is incorrect of rthat reason again, but more
- importantly, you did not set it up correctly!
- you need to express each function in terms of 'y'
- then use the "washer method"...also, your answer
- of infinity does not make sense!  You have a finite
- area and a finite region...the volume MUST be finite!

2. You need to use the weight density (10 lbs/50 ft
-- = 0.2 lbs/ft) then determine the work required to
-- move a small "section" of the cable y (or 40-y) ft
-- and go from there.

I hope this helps...TTYL, Abe