Definite and indefinite integrals

Key Questions

  • How do I evaluate definite integrals?

    A definite integral looks like this:

    int_a^b f(x) dx

    Definite integrals differ from indefinite integrals because of the a lower limit and b upper limits.

    According to the first fundamental theorem of calculus, a definite integral can be evaluated if f(x) is continuous on [a,b] by:

    int_a^b f(x) dx =F(b)-F(a)

    If this notation is confusing, you can think of it in words as:

    The integral of a function (f(x)) with limits a and b is the integral of that function evaluated at the upper limit (F(b)) minus the integral of that function evaluated at the lower limit (F(a))
    F(x) just denotes the integral of the function.

    Note that you will get a number and not a function when evaluating definite integrals. Also, you have to check whether the integral is defined at the given interval.

    Let's look at an example .

    int_-2^6 x^3+2 dx

    x^3+2 is defined for all real numbers, so the boundaries of a and b are defined. To evaluate this definite integral, we first find the integral function and then plug in the upper limit of 6 into the integral function, and subtract the integral function evaluated at the lower limit of -2.

    int_-2^6 x^3+2 dx = [1/4x^4+2x ]_-2^6=(1/4(6)^4+2(6))-(1/4(-2)^4+2(-2))= (336)-(0)= 336

    Sally · 1 · Oct 14 2014
  • How do I evaluate indefinite integrals?

    Indefinite integrals are antiderivatives in general form.

    int f(x) dx=F(x)+C,

    where F'(x)=f(x).

    I hope that this was helpful.

    Wataru · · Oct 19 2014
  • What is the difference between definite and indefinite integrals?

    Indefinite integrals have no lower/upper limits of integration. They are general antiderivatives, so they yield functions.

    int f(x) dx=F(x)+C,

    where F'(x)=f(x) and C is any constant.

    Definite integrals have lower and upper limits of integration (a and b). They yield values.

    int_a^b f(x) dx = F(b)-F(a),

    where F'(x)=f(x).

    I hope that this was helpful.

    Wataru · 4 · Oct 10 2014

Questions

Introduction to Integration