Definite Integrals

If f is defined on the closed interval [a, b] and the limit of the Riemann sums equation image indicator $LaTeX: \lim_{\|\Delta\| \to 0}\sum_{i=1}^{n}f(c_{1} )\Lambda x_{1}$ exists, then f is integrable on [a, b] and the limit is denoted by $LaTeX: \lim_{\|\Delta\| \to 0}\sum_{i=1}^{n}f(c_{1} )\Lambda x_{1} = \int_a^{b}f\left(x\right)dx$ . This limit of Riemann sums is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration. A definite integral is a number, unlike an indefinite integral, which is a family of functions.

All continuous functions are integrable. If a function f is continuous on an interval [a, b], then its definite integral over [a, b] exists. Note that many discontinuous functions are also integrable.

Definite Integrals 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 equation image indicator $LaTeX: A_a^b= \int_a^{b}f\left(x\right)dx$ . As is true with Riemann sums, any area below the x-axis is negative. If an integrable function y = f(x) has both positive and negative values on an interval [a, b], then the areas above the x-axis are added to the negatives to the negatives (opposites) of the areas below the x-axis.

Consider the area bounded by the function equation image indicator $LaTeX: f\left(x\right)=\cos x\:on\:\left[0,2\pi\right]$ . Integration from 0 to $LaTeX: 2\pi$ would produce 0, which does not accurately reflect the sum of the three areas.

cos x on graph with shaded areas

The integral video listed in the More Resources sidebar section may be helpful as it summarizes Riemann sums and provides a geometric interpretation of definite integrals as the area of a bounded region.

View the two presentations below introducing definite integrals and their application to distance, velocity, and area.

Properties of Definite Integrals

The definition of a definite integral over an interval [a, b] is based on a < b and moving along the interval from left to right. There are times when it is more convenient to move right to left or to consider the situation when a = b. The properties of definite integrals shown in the presentation below are very useful in computing definite integrals.

CLICK HERE to explore Wolfram|Alpha's widget: The Definite Integrator that provides self-check opportunities with a variety of self-entered problems.

Definite Integrals

1. Suppose that f and h are integrable and that equation image indicator $LaTeX: \int_1^{9}f\left(x\right)dx=-1, \int_7^{9}f\left(x\right)dx=5, \int_7^{9}h\left(x\right)dx=4$ . Evaluate the following integrals:

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