Using integration to determine the area under a curve

In summary, when trying to determine the flow rate of an air jet at 10D from the nozzle using a graph of velocity against vertical reading, you can use the trapezoid or simpsons rule to approximate the area under the curve and then calculate the volume flow rate using this area and the velocity.
  • #1
CaspianTiger
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Hi, i am doing some course work where i have to determine the flow rate of an air jet at 10D from the nozzle.


Basically I have a graph of velocity against the vertical reading from a manometer and i need to determine volume flow rate by finding the area under this curve. Unfuortunetly i do not understand this technique i am led to believe the trapezoidal or simpsons rule can be used to do this.

The volume flow rate is Q=VA

and the integration used i believe is A=2 [tex]pi[/tex] [tex]\int[/tex] V rdr

I have looked in my books but this only leads to the confusion. The book suggests i split up all the parts into rectangles and then do the integration for each rectangle then sum them, however using the technique in the book i think all the results will be the same. I do not udnerstand the explanation of the trapezoidal rule and i am not sure how i would apply simpsons rule to this.

This may appear simple but it has me very confused.


Thanks for the help.
 
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  • #2
The trapezoid rule is a numerical integration technique used to approximate the area under a curve when the exact area is unknown. It works by taking the average of two adjacent points on the curve, and then multiplying it by the change in the independent variable (in this case, the vertical reading). This produces an estimate of the area that is accurate for linear functions, and is generally reasonably accurate for more complicated functions as well. Simpson's rule is another numerical integration technique that is more accurate than the trapezoid rule. It works by taking the average of three adjacent points on the curve, and then multiplying it by the change in the independent variable. This produces an estimate of the area that is more accurate than the trapezoid rule. In both cases, you will need to calculate the change in the independent variable (i.e. the vertical reading) for each point on the graph, and then use the appropriate rule to calculate the area under the curve. Once you have the area, you can then calculate the volume flow rate by multiplying the area by the velocity of the air jet at 10D from the nozzle.
 

What is integration?

Integration is a mathematical process of finding the area under a curve by dividing it into smaller, simpler shapes and then summing up their areas.

Why is integration used to determine the area under a curve?

Integration is used because it allows for precise calculation of the area under a curve, even if the curve is complex or irregular.

What is the difference between definite and indefinite integration?

Definite integration involves finding the area under a curve between specific limits, while indefinite integration involves finding an equation for the curve itself.

How does the shape of the curve affect the integration process?

The shape of the curve affects the integration process by determining the type of integration method that should be used. For example, a curved line may require a different method than a straight line.

What are some real-world applications of using integration to determine the area under a curve?

Integration is commonly used in physics, engineering, economics, and other fields to calculate quantities such as work, volume, and profit that are represented by curves in real-world scenarios.

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