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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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

$y = e^{\frac{x}{2}} + C$ $y = e^(x/2) + C$ would be a good example, where $C$ $C$ is a constant.

Explanation:

I'm assuming we have

$\int_{0}^{1} \sqrt{1 + \frac{1}{4} e^{x}} d x$ $int_0^1 sqrt(1 + 1/4e^x)dx$

We can compare to the formula for arc length

$\int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$ $int_a^b sqrt(1 + (dy/dx)^2) dx$

So we have that $\frac{d y}{d x} = \sqrt{\frac{1}{4} e^{x}} = \frac{1}{2} \sqrt{e^{x}} = \frac{1}{2} {(e^{x})}^{\frac{1}{2}} = \frac{1}{2} e^{\frac{x}{2}}$ $dy/dx = sqrt(1/4e^x) = 1/2sqrt(e^x) = 1/2(e^x)^(1/2) = 1/2e^(x/2)$

We now integrate.

$y = e^{\frac{x}{2}} + C$ $y = e^(x/2) + C$

Thus an example of the curve would be $y = e^{\frac{x}{2}} + C$ $y = e^(x/2) + C$ .

Hopefully this helps!

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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?

1 Answer

Explanation:

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