Two functions having the same areas x to dx

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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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Hootenanny
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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.
 
arildno
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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.
 

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