Integrating Graphs with Two Intervals: How to Find the Mean?

  • Context: Undergrad 
  • Thread starter Thread starter Dollydaggerxo
  • Start date Start date
  • Tags Tags
    Graphs intervals
Click For Summary
SUMMARY

This discussion focuses on integrating a function f(x) defined as k within two intervals, specifically a < x < b and c < x < d, while being zero outside these intervals. The integral of f(x) is calculated as the sum of the integrals over the specified intervals, resulting in the expression k(b - a) + k(d - c). To find the mean value of f(x) over these intervals, one must divide the total integral by the combined length of the intervals, which is b - a + d - c.

PREREQUISITES
  • Understanding of definite integrals
  • Familiarity with the concept of mean value in calculus
  • Knowledge of piecewise functions
  • Basic skills in mathematical notation and integration techniques
NEXT STEPS
  • Study the properties of definite integrals in calculus
  • Learn about piecewise functions and their applications
  • Explore the concept of mean value theorem for integrals
  • Practice integrating functions with multiple intervals
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as educators looking to enhance their understanding of integration techniques involving piecewise functions.

Dollydaggerxo
Messages
60
Reaction score
0
In general,
if f(x)=k but only for a < x < b and c < x < d
how would you integrate the graph ?
To find the mean for example, would you find the mean for each and add together, or is there some special thing for this case?
 
Physics news on Phys.org
What's the value of f(x) for x not in the two intervals? Presumably the value is zero, but you didn't state this.

Assuming for the moment that f(x) = 0 outside the two intervals, finding the integral of this function (not the integral of its graph) is easy to find.
[tex]\int_{-\infty}^{\infty} f(x) dx~=~\int_a^b f(x) dx~+~\int_c^d f(x) dx~=~k(b - a)+ k(d - c)[/tex]

To find the mean value of this function over the two intervals, evaluate the integral of the function (see above), and then divide by the length of the two intervals, b - a + d - c.
 
sorry, yes I meant if it is zero everywhere else.
Thanks
 

Similar threads

Undergrad Open interval or Closed interval in defining convex function
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
5K
MHB 2.2.212 AP Calculus Exam problem find increasing interval
  • · Replies 2 ·
Replies
2
Views
2K
MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines
  • · Replies 3 ·
Replies
3
Views
2K
High School Explain Integration to me, please
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad The Basic Area Problem (introduction to the topic of integrals)
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Showing That a Function Does Not Have Two Distinct Roots
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Uniform convergence and derivatives -- difference between two theorems?
  • · Replies 1 ·
Replies
1
Views
3K
High School Finding intervals of a 3 degree function?
  • · Replies 4 ·
Replies
4
Views
2K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K