How are Integrals and Derivatives Related?

George E. Hrabovsky, President, MAST

Last time we learned about the derivative. This continues a basic survey of the principles of calculus.

A Google search of the word"integrals" brings up several words that could be different types of integrals. I also used MathWorld [1]. Here is the list:

• Definite Integral
• Indefinite Integral
• Elliptic Integral
• Multiple Integral
• Line Integral
• Improper Integral
• Surface Integral
• Volume Integral
• Path Integral
• Abelian Integral
• Darboux Integral
• Lebesgue Integral
• Denjoy Integral
• Euler Integral
• Haar Integral
• HK Integral
• Lebesgue-Stieltjes Integral
• Repeated Integral
• Fractional Integral
• Stieltjes Integral

We have already encountered the definite integral.

61. What is an indefinite integral?

62. What is an elliptic integral?

63. What is a multiple integral?

64. What is a line integral?

65. What is an improper integral?

66. What is a surface integral?

67. What is a volume integral?

68. What is a path integral?

69. What is an abelian integral?

70. What is a Darboux integral?

71. What is a Lebesgue integral?

72. What is a Denjoy integral?

73. What is an Euler integral?

74. What is a Haar integral?

75. What is an HK integral?

76. What is a Lebesgue-Stieltjes integral?

77. What is a repeated integral?

78. What is a fractional integral?

79. What is a Stieltjes integral?

We begin by using the same source as the last column [2]. The description of an indefinite integral requires a little bit of rewriting to become clear. If we assume that is the derivative of some function of , say then we can call this function the antiderivative of . So, since we know that

we know that the antiderivative is

Another way of writing this is

We can now write

where is some constant (since the derivative of a constant is always 0, this does nothing to change ). This is called the indefinite integral.  Taking the indefinite integral of something is just another way of asking the question, "If the function under study is a derivative, what was the function before the derivative was calculated?"

We can establish two rules that are relevant.

The First Fundamental Theorem of Calculus

If

and is smooth over the closed interval , then

This establishes that a definite integral may be calculated as the difference of two indefinite integrals.

The Second Fundamental Theorem of Calculus

If, is a smooth function defined on any open interval , and is some point within that interval, and we have

Then

These two theorems inextricably link derivatives to integrals as inverse operations.

[1] Eric W. Weisstein, MathWorld--A Wolfram Research Resource, http://mathworld.wolfram.com/.

[2] Paul Dawkins, "Complete Calculus I," http://tutorial.math.lamar.edu/, 2004.

Created by Mathematica  (August 1, 2005)