Home > Calculus > Determining the Length of a Curve

What is the arclength of #f(x)=x^3-xe^x# on #x in [-1,0]#?

Answer 1

Aurora Alvarado

Arc length #s=1.54116# units

The formula to determine length of arc s: #s=int_a^bsqrt(1+f' (x)^2) dx# #a=-1# and #b=0# the limits

#f(x)=x^3-x e^x#

#f' (x) 3 x^2 -[x*e^x*1+1*e^x]# #f' (x) = 3x^2 - x e^x - e^x#

#s=int_-1^0 sqrt(1+(3x^2 - x e^x - e^x)^2) dx#

There is no simple formula to evaluate the integral, so try using Simpson's Rule:

#s= 1.54116# units

Just observe the graph from #x=-1# to #x=0# graph{y=x^3-x e^x [-2.5, 2.5, -1.25, 1.25]}

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Answer 2

Paisley Johnson

To find the arc length of the function (f(x) = x^3 - xe^x) on the interval ([-1, 0]), you need to use the formula for arc length:

[L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx]

Where (f'(x)) represents the derivative of (f(x)).

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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.