Question

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?

Answer 1

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

Explanation:

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!