Find the area beween the curves y=x^2 and x+y=2 and the x axis

  • Context: MHB 
  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Area Axis Curves
Click For Summary

Discussion Overview

The discussion revolves around finding the area between the curves defined by the equations \(y=x^2\) and \(x+y=2\), as well as the x-axis. Participants explore the intersections of these curves and the method for calculating the area between them, considering different intervals and approaches.

Discussion Character

  • Mathematical reasoning, Debate/contested, Homework-related

Main Points Raised

  • One participant questions the relevance of the x-axis in the area calculation, suggesting it may be outside the area of interest.
  • Another participant asks how to find the area between each curve and the x-axis individually and inquires about the intersection points of the curves.
  • Several participants agree on setting the equations equal to each other to find intersection points, identifying \(x=-2\) and \(x=1\) as the points of intersection.
  • One participant proposes a method of finding the area by calculating the area of the larger function and subtracting the area of the smaller function, using a pizza analogy to illustrate the concept.
  • A participant presents an integral setup for calculating the area between the curves, but expresses uncertainty about the result, indicating a value of \(\frac{9}{2}\).
  • Another participant suggests a different integral setup for a specific interval, leading to a value of \(-\frac{2}{3}\), which raises questions about the correctness of the approach.
  • Discrepancies in results are noted, with one participant unable to identify differences without seeing another's work.
  • A later reply acknowledges a misunderstanding regarding the area being discussed and clarifies the correct interpretation of the region involved.

Areas of Agreement / Disagreement

Participants express differing views on the correct setup and calculation of the area, with no consensus reached on the final area value or the method used. Multiple competing approaches and interpretations remain present in the discussion.

Contextual Notes

Some participants' calculations appear to depend on specific intervals, and there are unresolved questions regarding the setup of integrals and the interpretation of the area to be calculated.

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
Find the area beween the curves $y=x^2$ and $x+y=2$ and the x axis

First on graphing these the $x-axis$ seem irrelevant in that it is outside the area to find.

[desmos="-10,10,-10,10"]y=x^2;y=-x+2[/desmos]
 
Physics news on Phys.org
How would we find the area between each curve and the x-axis individually?

Where do these curves intersect?
 
Set the equations equal to each other.
$x=-2$ $x=1$
 
karush said:
Set the equations equal to each other.
$x=-2$ $x=1$

Good! Ok, so now to find the area between them we need to find the area of the bigger part and then subtract out the area of of the smaller part. Just like if we had a whole pizza with a piece missing, we could take the area of the whole pizza minus the area of one slice to find the area of the whole pizza with a piece missing.

In this problem, which function has a larger area or put another way, which function is larger than the other over the interval?
 
$\int_{-2 }^{1 } (-x^2 - x+2)\,dx$
$\left[-\frac{{x}^{3}}{2 }-\frac{x}{2}+2x \right]^1_{-2}=\frac{9}{2}$

I hope
 
I am thinking, given the original problem statement, that you are to find the following area:

View attachment 4411
 

Attachments

  • karush3.png
    karush3.png
    2.6 KB · Views: 141
For the interval $0\le x\le2$ I get $-\frac{2}{3}$
 
Did you set it up as:

$$A=\int_0^1 x^2\,dx+\int_1^2 2-x\,dx$$ ?
 
No that would be 2/3
 
  • #10
I get a different result, but I can't tell where you and I differ without seeing your work...:D
 
  • #11
Yes, you are correct Mark. Sorry for misreading the question karush. I was thinking of this region in light green but Mark is correct. :)

[desmos="-10,10,-10,10"]y<x^2\left\{-2<x<1\right\};x+y<2\left\{-2<x<1\right\};[/desmos]
 
  • #12
No problem, how did you get the shading in demos?
 

Similar threads

High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Find the area enclosed by the curve y = x csch(x+y)
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Negative area above x-axis from integrating x^2?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad How to evaluate the enclosed area of this implicit curve?
  • · Replies 2 ·
Replies
2
Views
3K
High School Question about units for "area under curve"
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Inverse Functions: x=f(y) and X=f^-1(y)
  • · Replies 4 ·
Replies
4
Views
2K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K
High School Integration from "Area Under Curve" Perspective: Explained
  • · Replies 11 ·
Replies
11
Views
2K
MHB Find maximum area of a triangle
  • · Replies 1 ·
Replies
1
Views
1K
MHB Find the area between the curve and the x-axis
  • · Replies 2 ·
Replies
2
Views
2K