1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Limits of Integration

  1. Jan 22, 2014 #1
    1. The problem statement, all variables and given/known data

    Let f be the function defined by $$ f(x) = - ln(x) for 0 < x ≤ 1. $$ R is the region between the graph of f and the x-axis.


    b. Determine whether the solid generated by revolving region R about the y-axis has finite volume. If so, find the volume. If not, explain why.

    2. Relevant equations

    $$ y = -ln(x) $$
    $$ x = e^{-y} $$

    3. The attempt at a solution

    $$V = \pi \int_{x=0^+}^{x=1} [e^{-y}]^2 dy $$
    $$V = \pi\int_{∞}^{0} [e^{-2y}] dy $$
    $$\uparrow$$ This is my mistake.
    $$V = -\frac{\pi}{2}$$

    The actual solution is $$V = \pi \int_{0}^{∞}[e^{-2y}] dy = \frac{\pi}{2}$$
    But why are the limits of integration flipped? For part a, (Determine whether region R has a finite area. If so, find the area. If not, explain why.) my limits of integration were [x=0,x=1] $$ \int_{0^+}^{1} -ln(x) dx = 1 $$, so wouldn't I just set $$ e^{-y} $$ equal to 0 and 1 for part b? If not, please explain analytically why I need to flip my limits of integration, I can see from the graph that when x → 0, y → ∞ so please explain the problem analytically. The rotations about the y axes are very tricky for me and ANY advice would help :-) This is a very simple, but confusing concept.
    Last edited: Jan 22, 2014
  2. jcsd
  3. Jan 22, 2014 #2
    Essentially, since ##y=-\ln x## is decreasing on ##(0,1]##, ##dy\sim-dx##. I would advise either setting up the integral entirely in ##x## and then making an obvious (?) substitution, or just set it up entirely in ##y## from the get-go. when you mix stuff up like you did (integrand in one variable, limits in another) in ends badly more often than not in my experience.
  4. Jan 22, 2014 #3
    I just talked to my teacher and she told me that writing the coordinates are helpful when dealing with these type of problems. In this case they will be M(0,∞) and N(1,0) therefore, since we integrate with respect to x from left to right, my limits with respect to x will be $$ \int_{0}^{1} f(x)dx $$ and since we integrate with respect to y from bottom to top, my limits with respect to y will be $$ \int_{0}^{∞} g(y)dy $$
  5. Jan 22, 2014 #4
    Yes. This is what I mean when I say to set things up entirely in one variable or the other. It's just much easier that way.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted