I - Definite Integrals Lesson

Definite Integrals

If f is defined on the closed interval [a, b] and the limit of the Riemann sums $\lim_{\|\Delta\| \to 0}\sum_{i=1}^{n}f(c_{1} )\Lambda x_{1}$ exists, then f is integrable on [a, b] and the limit is denoted by $\lim_{\|\Delta\| \to 0}\sum_{i=1}^{n}f(c_{1} )\Lambda x_{1} = \int_a^{b}f\left(x\right)dx$. 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 $A_a^b= \int_a^{b}f\left(x\right)dx$. 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 $f\left(x\right)=\cos x\:on\:\left[0,2\pi\right]$. Integration from 0 to $2\pi$ would produce 0, which does not accurately reflect the sum of the three 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 Links to an external site.: 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 $\int_1^{9}f\left(x\right)dx=-1, \int_7^{9}f\left(x\right)dx=5, \int_7^{9}h\left(x\right)dx=4$. Evaluate the following integrals:

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