
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 1st, 2017, 05:19 AM  #1 
Newbie Joined: Jul 2017 From: Earth Posts: 6 Thanks: 1  Help explain curves which are entirely below the xaxis
Can someone help explain to me or maybe show me how to find the area of a curve which is below the xaxis. If the answer is a negative, is it correct ?
Last edited by JoKo; August 1st, 2017 at 05:22 AM. 
August 1st, 2017, 05:39 AM  #2  
Senior Member Joined: Jun 2015 From: England Posts: 676 Thanks: 194  Quote:
Curves don't have areas. They have length, curvature and slope. An area can be defined between a curve and another line. The simplest area is if that other line is one of the axes. We call the area between the x axis and the curve, "The area under the curve" Since this question is asked in calculus, I can say that this area corresponds to the definite integral between points (limits) on the same side of the axis as each other. Integrals below the x axis are counted (work out) as negative Integrals above the x axis are counted (work out) as positive BUT Areas are always positive For this reason the definite integral may be different from the total area under the curve. So the integral of a sine curve from 0 to 360 is zero, but the area under a sine curve is twice the integral of the sine from 0 to 180. http://www.bbc.co.uk/education/guide...gk7/revision/1 Does his help ? Last edited by studiot; August 1st, 2017 at 05:45 AM.  
August 1st, 2017, 05:41 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,037 Thanks: 1394 
I doubt that JoKo meant the area of a curve. The area bounded by various curves or lines was probably intended. Such an area can't be negative, regardless of whether or not the xaxis is part of the boundary.

August 1st, 2017, 07:10 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,720 Thanks: 699 
For example: find the area of the region bounded by the xaxis, y= 0, and the parabola, $\displaystyle y= x^2 1$. The first thing you should do is draw the graph or at least imagine it to see that the two cross at x= 1 and x= 1, that the only region bounded by those is between x= 1 and x= 1, and that, in that region, the xaxis is always above the parabola. To find the area, subtract the equation of the lower curve from the equation of the higher curve, 0 (x^2 1)= 1 x^2, and integrate from x= 1 to x= 1: $\displaystyle \int_{1}^1 1 x^2 dx= $$\displaystyle \left[ x \frac{x^3}{3}\right]_{1}^1=$$\displaystyle (1 \frac{1}{3}) (1+ \frac{1}{3})$$\displaystyle = 2 \frac{2}{3}= \frac{4}{3}$. Of course, "area" of any region is a positive number and we guaranteed that by subtracting "the equation of the lower curve from the equation of the higher curve". Last edited by Country Boy; August 1st, 2017 at 07:24 AM. 
August 1st, 2017, 08:06 AM  #5 
Senior Member Joined: May 2016 From: USA Posts: 802 Thanks: 318 
I have thanked so many posts here because I have learned something from this interesting question. I must admit that I had always interpreted the "area under the curve" as meaning a number that was positive if the area was above the xaxis and negative if below the xaxis. That is, I was using the term in a figurative or perhaps conventional sense rather than a literal sense. Obviously an area must be positive, but a limit can be positive, negative, or zero. So thanks to everybody. 

Tags 
curves, explain, xaxis 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Why does the other x axis decrease about one in this example?  afk0901  Calculus  8  May 4th, 2017 02:57 PM 
principal axis  Kinroh  Physics  1  August 6th, 2015 01:36 AM 
x axis and y axis Product of slopes is equal to 1  brhum  PreCalculus  4  November 27th, 2014 04:16 AM 
volume xaxis vs yaxis  thesheepdog  Calculus  2  November 10th, 2014 11:23 AM 
Caustic curves and dual curves  mdoni  Applied Math  0  February 18th, 2011 01:39 PM 