Indefinite integral and average distance

Click For Summary
SUMMARY

The discussion focuses on calculating the average distance to the x-axis for points in the region bounded by the x-axis and the graph of the function y = x - x². To find this average distance, one must integrate the function representing the distance to the x-axis over the defined region and divide by the area of that region. The area can be determined through integration, specifically by evaluating the integral of the function from the intersection points with the x-axis. The participants confirm that the region is indeed bounded by an upside-down parabola.

PREREQUISITES
  • Understanding of definite integrals and their applications
  • Familiarity with the concept of average value of a function
  • Knowledge of calculating areas under curves
  • Basic skills in sketching and interpreting graphs of functions
NEXT STEPS
  • Learn how to calculate the area under a curve using definite integrals
  • Study the average value theorem for integrals
  • Explore the properties of parabolic functions and their graphs
  • Practice solving problems involving integration of multivariable functions
USEFUL FOR

Students studying calculus, particularly those focusing on integration and applications of definite integrals, as well as educators teaching these concepts in a mathematical context.

-EquinoX-
Messages
561
Reaction score
1

Homework Statement



Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph of y = x - x^2.

Homework Equations





The Attempt at a Solution



Can someone guide me how to solve this?
 
Physics news on Phys.org
Pick any point in the region...what is the distance from that point to the x-axis?

If you had a finite number of points in the region, you would add up the distance from each point to the x-axis and divide by the number of points, correct?

Well, there are of course an infinite number of points in the region, so instead of adding, you integrate...
 
The "average value" of any function, f(x,y) on a region R is
\frac{\int_R f(x,y)dxdy}{area of R}
 
what is the area of R here? I already calculate the integral part
 
-EquinoX- said:
what is the area of R here? I already calculate the integral part

How do you usually calculate the area of a region?
 
How do you normally calculate the area of a region bounded by a curve and the x-axis? Have you made a sketch of the region?
 
yes I have made a sketch, it's an upside down parabola right? so I should do integration to find the area?
 
Yes, of course. You want
\frac{\int\int_R f(x,y)dydx}{\int\int_R dydx}[/itex]<br /> where f(x,y) is the &quot;distance to the x-axis&quot;.
 
okay got it. Thanks
 

Similar threads

The indefinite integral and its "argument"
  • · Replies 3 ·
Replies
3
Views
2K
Finding where a function is sign definite, sign indefinite or sign semidefinite
  • · Replies 2 ·
Replies
2
Views
1K
Expert Tips for Solving Indefinite Integral for Arc Length of f(x) = 3x^2
  • · Replies 11 ·
Replies
11
Views
2K
Triple Integral To Find Volume Between Cylinder And Sphere
  • · Replies 2 ·
Replies
2
Views
2K
How can the indefinite integral of e^(1/x)/(x(x+1)^2) be solved by hand?
  • · Replies 2 ·
Replies
2
Views
2K
Finding an indefinite integral
  • · Replies 4 ·
Replies
4
Views
2K
Please evaluate this double integral over rectangular bounds
  • · Replies 10 ·
Replies
10
Views
2K
Definite trigonometric integral using properties
  • · Replies 11 ·
Replies
11
Views
1K
Definite integral using only properties
  • · Replies 7 ·
Replies
7
Views
1K
The integral of a function ##f(x)## from its graph
  • · Replies 15 ·
Replies
15
Views
2K