Two functions having the same areas x to dx

In summary: Thanks for your question.In summary, the person is asking if there is a way to determine the value of variable A in the function f(x)=Ax so that it has the same area as a more complicated function, f(x)= x^2, when integrated between the same limits. The suggestion is to calculate both integrals and set them equal to solve for A.
  • #1
dmcoleman
1
0
Hi I have been thinking about an idea I have involving calculus that I think someone here can help me with. Is there a way you can determine a simple function (like f(x)=Ax) that has the same area dA from X to dX as another complicated function like f(x)=x^2. If you refer to the two attachments to this post you will see two graphs of two functions f(x)=X^2 and f(x)=Ax. The area under the curve of both functions from x=0 to x=2 are equal. My question is how can we determine the value of the variable A in the function f(x)=Ax if these conditions are to be satisfied. Thanks.
 

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  • #2
dmcoleman said:
Hi I have been thinking about an idea I have involving calculus that I think someone here can help me with. Is there a way you can determine a simple function (like f(x)=Ax) that has the same area dA from X to dX as another complicated function like f(x)=x^2. If you refer to the two attachments to this post you will see two graphs of two functions f(x)=X^2 and f(x)=Ax. The area under the curve of both functions from x=0 to x=2 are equal. My question is how can we determine the value of the variable A in the function f(x)=Ax if these conditions are to be satisfied. Thanks.
Welcome to PF,

Simple: Integrate both functions between the same two limits and set them equal. You should them be able to determine the value of your coefficient. Indefinite integration returns the area bounded by the curve, the limits and [in this case] the x-axis.
 
  • #3
1. Calculate each of the integrals, with "A" an as yet undetermined constant.
2. The requirement that these two integrals, i.e areas, gives you an equation you may solve for A.
 

1. What does it mean for two functions to have the same areas x to dx?

When two functions have the same areas x to dx, it means that the definite integrals of the two functions over the same interval [x, x + dx] have the same value. In other words, the area under the curve of both functions between the same two points is equal.

2. How can I determine if two functions have the same areas x to dx?

To determine if two functions have the same areas x to dx, you can evaluate the definite integrals of both functions over the same interval [x, x + dx]. If the values of the two integrals are equal, then the areas are the same.

3. Why is it important to know if two functions have the same areas x to dx?

Knowing if two functions have the same areas x to dx is important because it can help us understand the relationship between the two functions. It can also be useful in solving problems involving optimization and finding the area between two curves.

4. Can two functions have the same areas x to dx but still be different?

Yes, it is possible for two functions to have the same areas x to dx but still be different. This means that although the areas under the curves between the same two points are equal, the overall shape and behavior of the two functions may still be different.

5. How does the size of dx affect the areas of two functions?

The size of dx affects the areas of two functions because it determines the width of the interval over which the definite integral is being evaluated. A smaller dx will result in a more accurate calculation of the area, but it may also require more computation. A larger dx may result in a less accurate calculation, but it will require less computation.

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