## The Definite Integral

Read this section to learn about the definite integral and its applications. Work through practice problems 1-6.

### Definition of The Definite Integral

Each particular Riemann sum depends on several things: the function , the interval , the partition of the interval, and the values chosen for in each subinterval. Fortunately, for most of the functions needed for applications, as the approximating rectangles get thinner (as the mesh of P approaches 0 and the number of subintervals gets bigger) the values of the Riemann sums approach the same value independently of the particular partition P and the points . For these functions, the LIMIT (as the mesh approaches 0) of the Riemann sums is the same number no matter how the are chosen.

This limit of the Riemann sums is the next big topic in calculus, the definite integral. Integrals arise throughout the rest of this book and in applications in almost every field that uses mathematics.

Definition: **The Definite Integral**

then is integrable on the interval .

The number is called the Definite Integral of on and is written .

The symbol is read as "the integral from to of eff of dee or "the integral from to of with respect to ". The name of each piece of the symbol is shown in Fig. 1.

**Example 1:** Describe the area between the graph of , the x–axis, and the vertical lines at and as a limit of Riemann sums and as a definite integral.

Solution:

(from Table 2 in Section 4.1).

**Practice 1:** Describe the area between the graph of , the x–axis, and the vertical lines at and as a limit of Riemann sums and as a definite integral.

**Example 2: **Using the idea of area, determine the values of

Solution:

(a) represents the area between the graph of the x–axis, and the vertical lines at 1 and 3 (Fig. 2), and this area equals 6 square units.

b. represents the area between , the x–axis and the vertical lines at 0 and 4, so the integral equals 12 square units.

c. represents the area of 1/2 of the circle with radius 1 and center at (0,0), and the integral equals .

**Practice 2: **Using the area idea, determine the values of

**Example 3: **Represent the limit of each Riemann sum as a definite integral.

**Example 4:** Represent each shaded area in Fig. 3 as a definite integral. (Do not evaluate the definite integral, just translate the picture into symbols.)

The value of a definite integral depends only on the function being integrated and on the interval . The variable is a "dummy variable" and replacing it with another variable does not change the value of the integral. The following integrals each represent the integral of the function f on the interval , and they are all equal:

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.3-Definite-Integral.pdf

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