Chapter 8 Integral Calculus

Section 8.2 Definite Integral

8.2.3 Calculation Rules

Partition of the Interval of Integration 8.2.5
Let f:[a;b] be an integrable function. Then for every number z between a and b, we have

a b f(x)dx= a z f(x)dx+ z b f(x)dx.

With the definition

d c f(x)dx:=- c d f(x)dx

the rule above applies to all real numbers z for which the two integrals on the right-hand side of the equation exist, even if z does not lie between a and b. Before we demonstrate this calculation with an example, we will examine the definition above in more detail.
Exchanging the Limits of Integration 8.2.6
Let f:[a;b] be an integrable function. The integral of the function f from a to b is calculated according to the rule

b a f(x)dx=- a b f(x)dx.

The calculation rule described above is convenient when integrating functions that involve absolute values, or piecewise-defined functions.
Example 8.2.7
The integral of the function f:[-4;6],x|x| is

-4 6 |x|dx = -4 0 (-x)dx+ 0 6 xdx = [- 1 2 x2 ]-4 0 + [ 1 2 x2 ]0 6 = (0-(-8))+(18-0) = 26.

The integration over a sum of two functions can also be split up into two integrals:
Sum and Constant Multiple Rule 8.2.8
Let f and g be integrable functions on [a;b], and let r be a real number. Then

a b (f(x)+g(x))dx= a b f(x)dx+ a b g(x)dx. (8.2.2)

For constant multiples of a function, we have

a b r·f(x)dx=r· a b f(x)dx. (8.2.3)