How do you find the arc length of the curve #y = 2x - 3#, #-2 ≤ x ≤ 1#?

1 Answer
Bill K.
Jun 26, 2015

The arc length is #3sqrt{5}\approx 6.7082#

Explanation:

Since the graph of #y=f(x)=2x-3# is a straight line, there's actually no need to use calculus. Instead, just find the straight-line distance between the points #(-2,f(-2))=(-2,-7)# and #(1,f(1))=(1,-1)#. The answer, by the distance formula (Pythagorean theorem), is

#sqrt{(-2-1)^2+(-7-(-1))^2}=sqrt{9+36}=sqrt{45}#

#=sqrt{9}sqrt{5}=3sqrt{5}\approx 6.7082#

If you want to confirm this with calculus, evaluate the integral #int_{-2}^{1}sqrt{1+(f'(x))^2}\ dx=int_{-2}^{1}sqrt{1+4}\ dx#

#=sqrt{5}x|_{-2}^{1}=sqrt{5}(1-(-2))=3sqrt{5}#

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