Areas under curve and averages

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In summary, the problem at hand involves finding the area of a curve using calculus terms. The formula for finding the area is A = ∫(b-a)f(x)dx, where f(x) is the equation of the curve and b and a are the upper and lower limits of integration, respectively. This can be used to solve for the area of the curve in question.
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johnnyamerica
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This is a problem I'm working on by myself and I'm not sure how to go about it because I do not know how to phrase the question in calculus terms.

Imagine you have a simple quadrilateral between x=1 and x=4. A line passes from 1,2 to 4,5. The average height of the two points or of the line is 3.5, and the width of the shape is 3. The area when multiplying the average height by the width would be 10.5.

Now what if, instead of having a straight line from 1,2 to 4,5 there were a curve? I've taken the heights between the two closest points, averaged them and multiplied them by the width. I can only do this with two points (I can't take the average of three heights and multiply them by the width between the three points).

With a graph, I would have sets of two different heights at a time, take the average and multiply them by the width between, move on to the second & third point, third & fourth point, fourth & fifth and so on. I would then add up all the areas.

With points on a curve I'm not sure how to go about phrasing this because there would be an infinite number of different heights.

I'm just looking for any hints or suggestions right now. I'll continue to work on the problem but directing me to any similar problems will be very helpful.
 
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Thanks.To find the area of a curve, you can use the formula A = ∫(b-a)f(x)dx, where f(x) is the equation of the curve, b is the upper limit of integration, and a is the lower limit of integration. In this case, it would be A = ∫(4-1)f(x)dx. You can then solve the integral to find the area of the curve.
 

What is the purpose of calculating "Areas under curve and averages"?

The primary purpose of calculating "Areas under curve and averages" is to analyze and understand data in a visual and quantitative way. This allows scientists to make predictions and draw conclusions about trends and patterns in the data.

What is the difference between area under a curve and average?

The area under a curve represents the total value of a variable over a specific range, while the average represents the central tendency or typical value of the data. The average is calculated by dividing the sum of all values by the number of values, while the area under a curve is calculated by finding the integral of the curve over a given range.

How do you calculate the area under a curve?

The area under a curve can be calculated by finding the integral of the curve over a given range. This involves breaking the curve into smaller, more manageable sections and using mathematical techniques such as the trapezoidal rule or Simpson's rule to find the total area.

What does the area under a curve tell us about the data?

The area under a curve can provide important insights into the data, such as the total value of a variable over a specific range, the rate of change or growth, and the relationship between different variables. It can also help identify outliers or anomalies in the data.

How can "Areas under curve and averages" be applied in scientific research?

"Areas under curve and averages" can be applied in a variety of ways in scientific research. It can be used to analyze and compare data from experiments, track changes over time, and make predictions about future trends. It can also be used to support or refute hypotheses and provide evidence for scientific theories.

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