How do you find the length of the curve defined by #f(x) = x^2# on the x-interval (0, 3)?

Question

Cameron Alexander · Accepted Answer

The formula for the arc length for a curve is: #s = int_(a)^(b) sqrt(1 + ((dy)/(dx))^2)dx# which is a modified version of the distance formula implementing the limit definition of the derivative (zoom in very close, and the graph looks linear), so #Deltay# and #Deltax# are very small. This way, you accumulate very short straight lines along the curve to fit the curve. So, we need to find the derivative, and then square it: #((dy)/(dx))^2 = (2x)^2 = 4x^2# Plugging it in: #s = int_(0)^(3) sqrt(1 + 4x^2)dx# Assuming you are in your first year of Calculus, this is not something you might have been taught how to do. So maybe you should evaluate this on your calculator or on Wolfram Alpha numerically.  But, if you have heard of trigonometric substitution, that is what I would do. #= 2int_(0)^(3) sqrt(x^2 + 1/4)dx# With this, let:
#x = atantheta# where #a = sqrt(1/4) = 1/2#
#dx = 1/2sec^2thetad theta#
#sqrt(x^2 + 1/4) = sqrt(1/4tan^2theta + 1/4) = 1/2sectheta# So you now have the substituted integral (ignoring the bounds for now): #=> 2int 1/2sectheta*1/2sec^2thetad theta# #= 1/2int sec^3thetad theta# To do this difficult integral, we have to use integration by parts... #int udv = ?# Let:
#u = sectheta#
#dv = sec^2thetad theta#
#du = secthetatanthetad theta#
#v = tantheta# #= secthetatantheta - intsecthetatan^2thetad theta# #= secthetatantheta - intsectheta(sec^2theta - 1)dx# #= secthetatantheta - intsec^3thetad theta + int secthetad theta# #2int sec^3thetad theta = secthetatantheta + int secthetad theta# #int sec^3 thetadx = 1/2(secthetatantheta + ln|sectheta+tantheta|)# Thus: #= 1/2int sec^3thetad theta# #= 1/4(secthetatantheta + ln|sectheta+tantheta|)# Using the substitutions from before: #sectheta = sqrt(4x^2 + 1)#
#tantheta = 2x# #=> |[1/4(2sqrt(4x^2 + 1)x + ln(sqrt(4x^2 + 1) + 2x))]|_(0)^(3)# #= 1/4(2sqrt(4(3)^2 + 1)(3) + ln(sqrt(4(3)^2 + 1) + 2(3))) - cancel(1/4(2sqrt(4(0)^2 + 1)(0) + ln(sqrt(4(0)^2 + 1) + 2(0))))^(0)# #= color(blue)(1/4(6sqrt(37) + ln(sqrt(37) + 6)))# #~~ color(blue)("9.7471 u"^2)#

Evelyn Agard · Answer

undefined To find the length of the curve defined by f(x) = x^2 on the x-interval (0, 3), you can use the arc length formula:

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

where f'(x) is the derivative of the function f(x) with respect to x.

For the given function f(x) = x^2, its derivative is f'(x) = 2x.

So, substituting into the formula:

$$L = \int_{0}^{3} \sqrt{1 + (2x)^2} \, dx$$

$$L = \int_{0}^{3} \sqrt{1 + 4x^2} \, dx$$

You can then evaluate this integral to find the length of the curve.