Chapter 4: Integration
Section 4.2: The Definite Integral
Essentials
The value a Riemann sum takes as the number of subintervals becomes infinite is called the definite integral. For the sake of recall, this notion, formalized in Definition 4.2.1, might crudely be distilled to the phrase: "the definite integral is the limit of a Riemann sum."
Definition 4.2.1: The Definite Integral
For a function f continuous on the interval a,b, the limit of any of its Riemann sums exists, and all such Riemann sums have the same limit. This unique value is called the definite integral, and is denoted by the symbol
∫abfx ⅆx
The number at the bottom of the integral sign ∫ is called the lower limit of integration; the number at the top, the upper limit of integration. Thus, a would be the lower limit; and b, the upper. The function f is called the integrand, and the independent variable x is called the variable of integration.
Table 4.2.1 lists ten properties of the definite integral; where it appears, c is a constant.
∫aafx ⅆx=0
∫abc ⅆx=c b−a
∫abfx ± gx ⅆx=∫abfx ⅆx ± ∫abgx ⅆx
∫abc fx ⅆx=c ∫abfx ⅆx
∫abfx ⅆx=∫acfx ⅆx+∫cbfx ⅆx
∫abfx ⅆx= −∫bafx ⅆx
fx≥0 on a≤x≤b ⇒ ∫abfx ⅆx≥0
fx≥gx on a≤x≤b ⇒ ∫abfx ⅆx≥∫abgx ⅆx
m≤fx≤M on a≤x≤b ⇒ m b−a≤∫abfx ⅆx≤M b−a
∫abfx ⅆx≤ ∫abfx ⅆx
Table 4.2.1 Ten basic properties of the definite integral
Property (1) says that the area computed over an empty interval is zero.
Property (2) says that the area between the constant function y=c and the x-axis is the area of the rectangle of height c and width b−a.
Properties (3) and (4) say that the definite integral is a linear operator, just like the derivative operator. A linear operator is one that goes past constants and addition/subtraction.
Property (5) says that area under a curve is additive.
Property (6) says that reversing the limits of integration changes the sign of the definite integral.
Property (7) says that a positive function has positive area between its graph and the x-axis.
Property (8) says that if one function dominates another, the area beneath the one dominates the area beneath the other.
Property (9) bounds the area under a function by the areas of the rectangles whose heights are the maximum and minimum of the function.
Property (10) says that the absolute value of a definite integral is not more than the definite integral of the absolute value of the function. This is eminently reasonable: if the function has both positive and negative values, the portion below the x-axis contributes "negative area" to the sum of the areas of regions both above and below the x-axis. Necessarily, this sum is not more than the sum obtained by treating any area below the x-axis as positive, which is what happens when the absolute value of the function is integrated.
Examples
For the examples in Section 4.2, definite integrals will be evaluated by Maple. In Section 4.3, a mechanism by which definite integrals are evaluated will be studied.
Example 4.2.1
By evaluating an appropriate definite integral, obtain the area above the x-axis, but under the graph of fx=6+x−x2,x∈−2,3.
Example 4.2.2
Evaluate the definite integral of 2 sinx+3 cosx on the interval 0,π/2.
Verify that for this integrand the definite integral is a linear operator.
Example 4.2.3
Obtain the value of the definite integral ∫−12x3−7 x2+5 x+4 ⅆx as the limit of a right Riemann sum.
Verify the result by using Maple to evaluate the definite integral.
Example 4.2.4
Using fx=x3−7 x2+5 x+4 on the interval −1,2, verify Property (6) in Table 4.2.1.
Example 4.2.5
Calculate the area bounded by the x-axis, fx=x3−7 x2+5 x+4, and the lines x=−1,x=2.
Example 4.2.6
Graph the function Fx=∫−2xft ⅆt, where ft=t3−7 t2+5 t+4.
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