How do you find the length of the curve #y=e^x# between #0<=x<=1# ?

Answer 1
#L=int_0^1sqrt{1+({dy}/{dx})^2} dx#
#=int_0^1sqrt{1+e^{2x}}dx#
by the substitution #u=sqrt{1+e^{2x}}#.

#Rightarrow {du}/{dx}=e^{2x}/sqrt{1+e^{2x}}={u^2-1}/u Rightarrow dx={u}/{u^2-1}du#

As #x# goes from #0# to #1#, #u# goes from #sqrt{2}# to #sqrt{1+e^2}#
#=int_{sqrt{2}}^{sqrt{1+e^2}}u^2/{u^2-1} du#

by partial fraction decomposition,

#=int_{sqrt{2}}^{sqrt{1+e^2}}[1+1/2(1/{u-1}-1/{u+1})] du#
#=[u+1/2(ln|u-1|-ln|u+1|)]_{sqrt{2}}^{sqrt{1+e^2}}#
#=[u+1/2ln|{u-1}/{u+1}|]_{sqrt{2}}^{sqrt{1+e^2}}#
#=sqrt{1+e^2}+1/2 ln({sqrt{1+e^2}-1}/{sqrt{1+e^2}-1})-sqrt{2}-1/2ln({sqrt{2}-1}/{sqrt{2}+1})#

I hope that this was helpful.

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Answer 2

To find the length of the curve y = e^x between x = 0 and x = 1, you can use the arc length formula for a curve y = f(x) on the interval [a, b]:

Arc length = ∫[a, b] √(1 + (f'(x))^2) dx

For the given curve y = e^x on the interval [0, 1]:

f(x) = e^x f'(x) = e^x

Plug these into the arc length formula:

Arc length = ∫[0, 1] √(1 + (e^x)^2) dx

Now, integrate the expression:

Arc length = ∫[0, 1] √(1 + e^(2x)) dx

This integral can be difficult to evaluate analytically, so numerical methods or approximations may be necessary to find the precise value of the arc length.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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