Area between Curves: Find the Bounded Region

  • Thread starter Thread starter flyers
  • Start date Start date
  • Tags Tags
    Area Curves
Click For Summary

Homework Help Overview

The problem involves finding the area bounded by the curves defined by the equations x = 4 - y² and x = y² - 2y. The context is centered around understanding how to approach integration when the curves are expressed in terms of y rather than x.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants discuss the nature of the curves and their representation as functions of y. Some express uncertainty about integrating these curves, while others suggest that they can still be treated as functions for the purpose of finding the area. There are mentions of graphing the curves and determining intersection points as a starting point for the problem.

Discussion Status

The discussion is ongoing, with various participants offering insights into the nature of the curves and suggesting methods for approaching the problem. There is a recognition that the curves define a bounded region, and some participants emphasize the importance of graphing to visualize the area in question.

Contextual Notes

Participants note that the equations do not conform to the typical format of functions where y is the dependent variable, leading to some confusion about how to proceed with integration. There is an acknowledgment of the need to consider the curves in relation to the y-axis rather than the x-axis.

flyers
Messages
28
Reaction score
0

Homework Statement



Find the area bounded by the curves given by the graphs x = 4 - y^2 and x = y^2 - 2y


The Attempt at a Solution



I don't know how I can integrate these curves as they are not functions. Can someone tell me to get started on this problem?

Thank you
 
Physics news on Phys.org
flyers said:

Homework Statement



Find the area bounded by the curves given by the graphs x = 4 - y^2 and x = y^2 - 2y


The Attempt at a Solution



I don't know how I can integrate these curves as they are not functions. Can someone tell me to get started on this problem?

Thank you

They most certainly are functions. They just don't have x as the independent variable like you're used to. It matters little, you can still get it done.

Integrating is the same no matter the variable of integration...
 
flyers said:

Homework Statement



Find the area bounded by the curves given by the graphs x = 4 - y^2 and x = y^2 - 2y


The Attempt at a Solution



I don't know how I can integrate these curves as they are not functions. Can someone tell me to get started on this problem?

Thank you
They aren't functions, but they are curves, and they define a bounded region. Start by graphing them and finding where the two curves intersect, and then figure out what your typical area element looks like - i.e., horizontal strip of vertical strip- and its dimensions.
 
Mark44 said:
They aren't functions, but they are curves, and they define a bounded region. Start by graphing them and finding where the two curves intersect, and then figure out what your typical area element looks like - i.e., horizontal strip of vertical strip- and its dimensions.

Actually, they are functions. They're just functions of y, not the functions of x most people are used to.
 
you could also just solve for y with those two equations and you would have what you are use to. Just think of those as respect to the y-axis and not the x. So the region you are measuring is taking the integral of the y dy instead of x dx, still the same just a different reference.
 
Char. Limit said:
Actually, they are functions. They're just functions of y, not the functions of x most people are used to.
True, but people generally think of functions where y is the dependent variable and x is the independent variable, out of force of habit. In that sense, the equations don't represent functions.

In any case, the important thing is that the OP should graph both equations to find the region whose area is to be found.
 

Similar threads

Finding Area using parametric equation
  • · Replies 12 ·
Replies
12
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Find the minimum value of area under y = 4x - x^3
  • · Replies 7 ·
Replies
7
Views
2K
Simple integration to find area
  • · Replies 2 ·
Replies
2
Views
2K
Calculating Area Under a Curve Using Change of Variables and Quotient Rule
  • · Replies 5 ·
Replies
5
Views
2K
Area surface of revolution (rotating this astroid curve around the x-axis)
  • · Replies 2 ·
Replies
2
Views
2K
Area Between the Parametric Curves
  • · Replies 1 ·
Replies
1
Views
2K
Area under a curve-positive and negative area cancellation
  • · Replies 2 ·
Replies
2
Views
2K
Find the area delimited by two polar curves
  • · Replies 2 ·
Replies
2
Views
2K
Evaluating the area under a curve
  • · Replies 24 ·
Replies
24
Views
2K