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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: 915 Thanks: 271  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: 21,031 Thanks: 2262 
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: 3,264 Thanks: 902 
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: 1,310 Thanks: 552 
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. 

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curves, explain, xaxis 
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