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How do you find the arc length of the curve #y=e^(x^2)# over the interval [0,1]?

Answer 1

Emily Ammerman

#≈2.12762# numerical solution

#S = int_0^1 sqrt(1+(y')^2) dx#

# = int_0^1 sqrt(1+(2x e^(x^2))^2) dx#

#≈2.12762# numerical solution

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Answer 2

Christopher Agresta

To find the arc length of the curve ( y = e^{x^2} ) over the interval ([0,1]), follow these steps:

Determine the formula for arc length, which is given by ( \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ).
Find the derivative of ( y = e^{x^2} ) with respect to ( x ), which is ( y' = 2xe^{x^2} ).
Substitute the derivative into the formula for arc length.
Integrate ( \sqrt{1 + (2xe^{x^2})^2} ) over the interval ([0,1]).
Evaluate the integral to find the arc length of the curve over the given interval.

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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.