Definite integrals have many properties that make their calculation much simpler.
Most properties of definite integrals are also properties of limits since an integral is a limit of a summation, as shown below:
external image e1.gif where external image e2.gif is a Riemann Sum of f(x) on [a, b]

Properties of Definite Integrals

The Definite Integral as the Area of a Region

If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by:
  • Area=integral.JPG

Two Special Definite Integrals

If f is defined at x = a, then:
  • external image e14.gif

If f is integrable on [a, b], then:
  • external image e7.gif

Additive Integral Property

If f is integrable on the three closed intervals determined by a, b, and c, then:
  • external image e6.gif

Preservation of Inequality

If f and g are integrable and nonnegative on the closed interval [a, b] and f(x)g(x) for all x in [a, b], then:
  • external image e8.gif

Other Properties of Definite Integrals

If f is integrable on [a, b] and c is a constant, then the function c * f is integrable on [a, b], and:
  • external image e4.gif

If f and g are integrable on [a, b], then the function f + g is integrable on [a, b], and
  • external image e5.gif
  • Note: this property holds true with subtraction as well as addition, and also can be extended to cover any finite number of functions (such as: f(x) + g(x) + h(x) + ...)

Examples:


1. The integral of y = 3x + 4 from 2 to 2 is 0 because the integral of any function in which both bounds are the exact same number is zero.

2. If the integral from 1 to 4 of the function y = x² is 21, then the integral from 4 to 1 of x² is -21 because when the bounds of an integral are switched, the integral is the same, but has the opposite sign (positive changes to negative, or negative changes to positive).

3. If the integral of y = 2x + 3 from 1 to 3 is 14 and the integral of the same function from 3 to 5 equals 22, then the integral from 1 to 5 of the function y = 2x + 3 equals 14 + 22, or 36 because the two integrals (from 1 to 3 and 3 to 5) make up the whole integral (from 1 to 5). Conversely, since the integral of y = 4x³ - 2x from 2 to 4 equals 228, then you know that the sum of the integrals from 2 to 3 and 3 to 4 of the same function must also equal 228.

4. To calculate longer, more complicated integrals, you can take the original function, make each term a separate function, find the integral of each term, and then add them back together. In this example, the integral of y = 6x³ - 3x² from 1 to 3 can be rewritten as the integral of y = 6x³ from 1 to 3 plus the integral of y = -3x² from 1 to 3. Since the the integral of y = 6x³ from 1 to 3 equals 120 and the integral of y = -3x² from 1 to 3 equals -26, the integral of y = 6x³-3x² from 1 to 3 is 120 + -26, or 94.

Looking for another example of definite integrals?

Sources:

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus. 8th ed. Boston and New York: Houghton Mifflin Company, 2006. Pages: 271-281.

Husch, Lawrence S. "Definite Integrals." Visual Calculus. University of Tennessee. 16 May 2007. < http://archives.math.utk.edu/visual.calculus/4/definite.1/index.html >.