X
Explore It
Close

YOU WILL LEARN ABOUT:

Integrals of symmetric functions.

t30_concept_01 t30_concept_02

Sometimes, a function’s symmetry can make it easier to evaluate integrals of the function. If f(x) is an odd function, then ∫-aaf(x)dx=0 for any a. If fx is an even function, then ∫-aaf(x)dx=2∫0af(x)dx for any a, where it is usually easier to evaluate the antiderivative at 0 than at -a.

Even and odd functions arise in many situations. Symmetry allows integrals involving these functions to be evaluated more easily.

t30_when_img_01
Some animal scientists study interacting groups of predators and prey. The rates of change for the populations of such groups can be described by even and odd functions. The integrals of these functions give the net change in population.
t30_when_img_03
Particle physicists work with the wave functions of particles. If the wave function of a particle is even, then the function that describes the expected position of the particle is odd, and so in the long run, the particle is expected to be at the origin. This is because integrating an odd function with symmetric limits of integration always gives 0.
t30_when_img_02
Fresnel integrals are integrals of sint2 and cost2, both of which are even. They are most often used by lens designers, but they are also sometimes used by civil engineers when designing highways.

INTEGRALS OF SYMMETRIC FUNCTIONS
h(x)=xsin(x)
Also show x=-a
f(x)=px2+q
0
5
a=4.50
0
3
p=1.00
0
3
q=3.00
∫ 0 4.50 f ( x ) d x = 4.8750 ∫ - -4.50 0 f ( x ) d x = 4.8750 ∫ - -4.50 0 f ( x ) d x = - 4.8750 ∫ − 4.50 4.50 f ( x ) d x = 87.7500
Also show x=-a
g(x)=psin(qx)
0
5
a=5
0
3
p=3
0
3
q=3
∫ 0 2.50 g ( x ) d x = 4.8750
∫ - 2.50 0 g ( x ) d x = 4.8750
∫ 2.50 - 2.50 g ( x ) d x = 0.0000
Initial position of path    (    ,    )

Direction Field
dxdt=sinxcosy dydt=-cosxsiny
Also show x=-a
h(x)=xsin(x)
0
10
a=8.0
∫ 0 8 . 0 h ( x ) d x = 2.1534
∫ - 8 . 0 0 h ( x ) d x = 2.1534
∫ - 8 . 0 8 . 0 h ( x ) d x = 4.3067
Initial position of path    (    ,    )

Direction Field
dxdt=sinxcosy dydt=-cosxsiny

INSTRUCTIONS

Select a family of functions from the dropdown menu. Then adjust the parameters to see how they affect the values of the integrals.

PRACTICE QUESTIONS

1 2 3
1. For the function f ( x ) = 2 x 3 , which of the following equations is true?
2. For the function f ( x ) = 2.5 x 2 + 1 , which of the following equations is true?
3. For the function family f ( x ) = p x 3 , which parameters affect the value of ∫ − a a f ( x ) d x ?

INSTRUCTIONS

Select a family of functions from the dropdown menu. Then adjust the parameters to see how they affect the values of the integrals.

PRACTICE QUESTIONS

1 2 3
1. For the function g(x)=2sin(3x), which of the following shows the relationship between ∫02.5g(x)dx and ∫-2.50g(x)dx?
2. Consider the function g(x)=cos(2x). If ∫0ag(x)dx=-0.3784, what is the smallest possible positive value of a and what is the value of ∫-aag(x)dx?
3. For the function family g(x)=psin(qx), which of a, p, and q affect the value of ∫-aag(x)dx? Select all that apply.

INSTRUCTIONS

Select a function from the dropdown menu. Then adjust the parameter to see how it affects the values of the integrals.

PRACTICE QUESTIONS

1 2 3
1. If ∫-aah(x)dx=0 for all a, what could h(x) be? Select all that apply.
2. For which functions does ∫-aah(x)dx depend on the value of a? Select all that apply.
3. In which of the following circumstances does ∫-aah(x)dx=0? Select all that apply.
Result

Enter value between 0 to 6

Enter the value between -100 to 100

Enter the value between -10 to 10