The Definite Integral

Key Questions

  • How do I find a definite integral by computing an area?

    Answer:

    I don't have a one sentence answer

    Explanation:

    Above the x-axis
    If f(x) is non-negative over the interval [a,b], then
    int_a^b f(x) dx is the area of the region between the graph and the x-axis between x=a and x=b

    Below the x-axis
    If f(x) is non-positive over the interval [a,b], then
    int_a^b f(x) dx is the negative of area of the region between the graph and the x-axis between x=a and x=b

    Example:

    int_0^3 sqrt(9-x^2) dx.

    a=0 and b=3.
    f(x) = sqrt(9-x^2) is never negative, so it is not negative on [0,3]

    Therefore the integral is equal to the area under the curve and above the x-axis between x=0 and x=3.

    The graph of y = sqrt(9-x^2) is the part of y^2 = 9-x^2 that has non-negative y-values. It is the upper semicircle for x^2+y^2 = 9
    The part between x=0 and x=3 is a quarter of a circle with radius 3.

    graph{y = sqrt(9-x^2)*(sqrt(1.5^2-(x-1.5)^2))/(sqrt(1.5^2-(x-1.5)^2)) [-2.63, 6.137, -0.812, 3.572]}

    So
    int_0^3 sqrt(9-x^2) dx is 1/4 of the area of the circle with radius 3
    int_0^3 sqrt(9-x^2) dx = (9 pi)/4

    Jim H · 1 · Jun 19 2015
  • What is an indefinite integral?

    Answer:

    An indefinite integral of a function f(x) is a family of functions g(x) for which: g'(x)=f(x)

    Explanation:

    An indefinite integral of a function f(x) is a family of functions g(x) for which: g'(x)=f(x).

    Examples:

    1) if f(x)=x^3, then indefinite integral is:

    int x^3dx=x^4/4+C, because:

    (x^4/4)'=4*x^3/4=x^3, and C'=0 for any real constant C

    2) If f(x)=cosx, then

    int cosx dx= sinx+C, because:

    (sinx)'=cosx

    Daniel L. · 1 · Aug 31 2015
  • What is a definite integral?

    A definite integral is when you integrate a function over a specified interval. When completed you have a definite answer.

    Definite Integral because it is bounded

    int_0^1 3xdx evaluates to [(3(1)^2)/2 - (3(0)^2)/2] = 3/2 - 0 = 3/2=1.5

    Indefinite Integral because it is not bounded

    int 3x dx evaluates to (3x^2)/2 + C

    AJ Speller · 1 · Sep 10 2014

Questions

Limits, Motion, and Areas