Not understanding some integration results

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In summary, the equation seems to be a sine wave, but when integrated it gives a negative area until x = 0.56. This negative area might be because of a hidden constant.
  • #1
dBrandon/dC
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I was wondering if you guys could help me understand something. I have an equation that, when I graph it, looks a lot like a regular sine function. However, when I integrate it to find the area under it and graph the new function, I'm getting a negative result in the first 0.56 radians or so.

The equation is:

dy = xSIN(x) - 0.03356(x^2)SIN(x) dx

Integrating for x (actually, letting Wolfram Mathematica Online Integrator do it for me), I get:

y = (1 - 0.06712x)SIN(x) + 0.03356(x - 29.8643)(x + 0.0669695)COS(x)

(I'm trying to analyze this for x = 0 to pi, by the way.)

The first equation gives me nothing negative, so why when it's integrated and graphed do I get a negative dip before x = 0.56?
 
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  • #2
Unless you have a given initial condition for y (i.e. y(a) = b), you cannot graph the indefinite integral just knowing the area under the graph. Wolfram alpha gives you a + Constant at the end, not sure why you got rid of that. However, if the problem you're working on does have an initial condition, then you can use that to find the constant and graph the integral properly.
 
  • #3
Wolfram didn't mention a constant, though I know I could use one. I just didn't think I needed to. ... I'm including two pictures here - screenshots of an online graphing calculator (http://www.coolmath.com/graphit/) where I plugged in these equations. In both of them, I've zoomed into the region of x = 0 to about 3.14.View attachment Capture1.bmp View attachment Capture2.bmp The first picture is a graph of the original equation. As you can see, the curve is entirely in the y+ quadrant. But in the second picture, after the equation is integrated, the first part of the curve is negative. It's been a long time since I took a calculus class, but isn't an integration supposed to give me the area under the curve? And when I solve the integrated equation for, let's say, x = 0.5, shouldn't that give me the area under the curve down to the y = 0 line, and from x = 0 to x = 0.5? But when I solve for x = 0.5, I get -0.027. That makes sense given the second picture, but it doesn't make sense to me given the first picture. Where's the negative area under the curve in the first graph?
 
  • #4
I think you're getting definite integrals confused with indefinite integrals. If you're dealing with a definite integral (i.e. an integral with a lower bound and upper bound), you will get the exact area under the curve between the two x values. However, if you find the indefinite integral, you cannot graph it unless you know what the constant is. The program you used might have just set a default value for a constant, maybe 0.

Assuming you know what the constant is and looking at the two graphs you have, it's usually easier looking at the slope instead of area. The slope of the second graph should be the y-value of the first graph.
 
  • #5
Bingo. If I add a constant, bringing y up to 0 when x = 0, that solves my problem, and the rest of the numbers line up with what I thought they should be. Thanks a lot!
 

What is integration and why is it important in science?

Integration is the process of combining different pieces of information or data to form a more complete understanding or picture. In science, integration is crucial because it allows us to connect different concepts or findings and create a more comprehensive understanding of a specific topic or phenomenon.

Why do some integration results seem confusing or difficult to understand?

Integration can be a complex process, involving a variety of data, theories, and perspectives. It is not uncommon for integration results to be challenging to understand because they often require a deep understanding of the individual components and their relationships.

What can I do if I am having trouble understanding integration results?

If you are struggling to understand integration results, it may be helpful to break down the information into smaller pieces and examine each one individually. You can also reach out to experts in the field or consult additional resources to gain a better understanding.

How can I ensure that my integration results are accurate?

To ensure accuracy in integration results, it is essential to use reliable and valid data, consider multiple perspectives, and carefully evaluate the connections between different pieces of information. It can also be helpful to have other experts review your work for any potential errors or biases.

What are some common challenges in integrating different scientific findings or theories?

One of the most significant challenges in integration is finding a way to reconcile conflicting or contradictory data or theories. Additionally, integrating different types of data or perspectives can be challenging, as they may not align easily. It is also crucial to avoid oversimplification and reductionism when integrating complex information.

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