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!

Fundamental theorem of calculus

  1. Mar 3, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx## exist.

    2. Relevant equations

    None


    3. The attempt at a solution

    I am wondering how this was solved?

    Why is that from the fundamental theorem of calculus we get ##\frac{d}{dt}\int^t_a\left(\int^d_cf(x,y)dy\right)dx## ##=## ##\int^d_cf(t,y)dy## ?
     
  2. jcsd
  3. Mar 3, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    Let [itex]F: [a,b] \to \mathbb{R}: x \mapsto \int_c^d f(x,y)\,dy[/itex], and apply the fundamental theorem of calculus to [itex]\frac{d}{dt}\int_a^t F(x)\,dx[/itex].
     
  4. Mar 3, 2014 #3
    I am not sure what exactly to do. Can you elaborate further please?
     
  5. Mar 3, 2014 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you please cite what exactly you mean with "the fundamental theorem of calculus"?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Fundamental theorem of calculus
Loading...