Area Under a Curve: Calculations & Explanation

In summary: If dx means a small amount of x, then the integral will still be a small amount of x, only at which point? Well, if you graph x then dx in the graph must still be a small amount of x, only at which point does the area under the curve equal two? Unfortunately, this is not always the case. If the equation you are trying to solve has a linear or quadratic form, then the area under the curve may not be equal to two. So in summary, if you are trying to find the area under a curve y=x from 0 to 2, you should use the integral method.
  • #1
yyttr2
46
0
Hey everyone :D

I know basic integration and differentiation and I was siting on my bed today thinking how on Earth any of it could correlate to finding the area under a curve.
So! I got out my trusty pen and paper and got to work!

After much thought I wrote: " If [tex]dx[/tex] means 'a small amount of x' then, if we graph x then dx in the graph must still be a small amount of x, only... at which point?
So I just assumed that dx one part of x as a whole. Remembering some of my physics class I calculated the area under the graph of "x" from points 0-2 on the x-axis which equals 2.
So in my assumtion dx would be from 1-2 or 0-1 on the x axis.
and therefor.
the area under the line must equal: [tex]\int dx[/tex]
but, from 0-2 how do you calculate it? [tex]\int^{2}_{0}dx=[/tex] which does equal two.

So, If the applies for linear equations it must apply for quadratic equations as well, right?

So in the equation [tex]f(x)=-7x^{2}+6[/tex]
The area under the curve for this from 1 to 2 on the x-axis must equal:
[tex]\int^{2}_{0}(-7x^{2}+6)dx[/tex]
correct?
 
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  • #2
yyttr2 said:
After much thought I wrote: " If [tex]dx[/tex] means 'a small amount of x' then, if we graph x then dx in the graph must still be a small amount of x, only... at which point?
So I just assumed that dx one part of x as a whole. Remembering some of my physics class I calculated the area under the graph of "x" from points 0-2 on the x-axis which equals 2.
So in my assumtion dx would be from 1-2 or 0-1 on the x axis.
and therefor.
the area under the line must equal: [tex]\int dx[/tex]
but, from 0-2 how do you calculate it? [tex]\int^{2}_{0}dx=[/tex] which does equal two.

If you are finding the area under the line y=x from 0 to 2, the integral should be

[tex]\int_0 ^{2} x dx = 2[/tex]

When you draw the line y=x and draw the verticals x=0 and x=2, the distance Δx, represents a small strip if width 'Δx' and length 'y' . So the area of that strip is 'y Δx'. To find the total area, one sums up all the small strips, from x=0 to x=2. But for the area to be exact, the limit must be taken as Δx tends to 0.

[tex]A= \lim_{x \rightarrow 0} \sum_{x=0} ^{x=2}y Δx = \int_0 ^{2} y dx[/tex]

and in this case y=x. So the integral becomes

[tex]A= \int_0 ^{2} x dx[/tex]


Only in special cases can you find the area using geometric methods, for example, y=x and x=0,x=2, forms a triangle.
 
  • #3
When you draw the line y=x and draw the verticals x=0 and x=2, the distance Δx, represents a small strip if width 'Δx' and length 'y' . So the area of that strip is 'y Δx'. To find the total area, one sums up all the small strips, from x=0 to x=2. But for the area to be exact, the limit must be taken as Δx tends to 0.
[tex]A=lim_{x\rightarrow0}\sum^{x=2}_{x=0}y916;x=\int^{2}_{0}ydx [/tex]

Yes *cough* what you said.

and I was assuming [tex]\int^{2}_{0}dx[/tex] was the same as [tex]\int^{2}_{0}xdx[/tex]
Sorry for that mistake :)
 
  • #4
yyttr2 said:
I know basic integration and differentiation and I was siting on my bed today thinking how on Earth any of it could correlate to finding the area under a curve.
Because of the very definition of the (Riemann) integral.
 

Related to Area Under a Curve: Calculations & Explanation

1. What is the purpose of calculating the area under a curve?

The area under a curve can represent various aspects of a phenomenon or data set, such as the total quantity, the average value, or the rate of change over time. It can also be used to compare and analyze different sets of data or to make predictions.

2. How do you calculate the area under a curve?

The most common method is to use numerical integration, such as the trapezoidal rule or Simpson's rule, which involves breaking the curve into smaller segments and calculating the area of each segment. There are also analytical methods, such as the fundamental theorem of calculus, which can be used for certain types of curves.

3. What factors can affect the accuracy of the area under a curve calculation?

The accuracy of the calculation can be affected by the method used, the number of segments or intervals chosen, and the smoothness of the curve. The more segments used, the more accurate the calculation will be. Additionally, any errors in the data or assumptions made in the calculation can also affect the accuracy.

4. Can the area under a curve be negative?

Yes, the area under a curve can be negative if the curve dips below the x-axis. This indicates that the total quantity or value represented by the curve is decreasing over time or across the given interval.

5. How is the area under a curve used in real-world applications?

The concept of area under a curve is used in various fields, such as physics, economics, and engineering, to analyze and understand data. It can be used to calculate the work done by a variable force, the profit or loss of a business over time, or the amount of material needed for a construction project. It is also used in statistics to calculate probabilities and make predictions based on data trends.

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