The Fundamental Theorem of Calculus

Key Questions

  • How do you use the Fundamental Theorem of Calculus to evaluate an integral?

    If we can find the antiderivative function #F(x)# of the integrand #f(x)#, then the definite integral #int_a^b f(x)dx# can be determined by #F(b)-F(a)# provided that #f(x)# is continuous.

    We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that #f(x)# is continuous and why.

    FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.

    For most students, the proof does give any intuition of why this works or is true. But let's look at #s(t)=int_a^b v(t)dt#. We know that integrating the velocity function gives us a position function. So taking #s(b)-s(a)# results in a displacement.

    Amory W. · · Aug 20 2014
  • How do you use the Fundamental Theorem of Calculus to find the derivative of #f(x)=int_1^(x)sqrt(e^t+sin(t))dt# ?

    Fundamental Theorem of Calculus

    #d/{dx}int_a^x f(t) dt=f(x)#

    This theorem illustrates that differentiation can undo what has been done to #f# by integration.

    Let us now look at the posted question.

    #f'(x)=d/{dx}\int_1^xsqrt{e^t+sint}dt=sqrt{e^x+sinx}#

    I hope that this was helpful.

    Wataru · · Oct 9 2014
  • What is the Fundamental Theorem of Calculus for integrals?

    #int_a^b f(x) dx=F(b)-F(a)#,
    where F is an antiderivative of #f#

    Wataru · · Sep 9 2014

Questions

Introduction to Integration