Find a curve through the region in the XY-plane whose length from X=0 to X=1 is L=INTEGRATION sqrt 1+1/4e^xdx?

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Answer 1 · 2018-05-27T16:21:37

#y = e^(x/2) + C# would be a good example, where #C# is a constant.

I'm assuming we have

#int_0^1 sqrt(1 + 1/4e^x)dx#

We can compare to the formula for arc length

#int_a^b sqrt(1 + (dy/dx)^2) dx#

So we have that #dy/dx = sqrt(1/4e^x) = 1/2sqrt(e^x) = 1/2(e^x)^(1/2) = 1/2e^(x/2)#

We now integrate.

#y = e^(x/2) + C#

Thus an example of the curve would be #y = e^(x/2) + C#.

Hopefully this helps!