How do you use the Trapezoidal Rule with n=60 to estimate the length of the curve #y=sinx#, with x greater or equal to 0 and x less than or equal to pi?

1 Answer
Oct 10, 2015

Answer:

The length of the curve #y=f(x)# from #a# to #b# is #int_a^b sqrt(1-(f'(x))^2) dx#

Explanation:

So we need to approximate #int_0^pi sqrt(1-cos^2x) dx#

using the trapezoidal rule. We'll use the formula.

#T_60 = 1/2 Deltax(f(x_0)+2f(x_1)+2f(x_3)+* * * +2f(x_(n-1)+f(x_n))#

In this case #f(x) = sqrt(1-cos^2x)dx#

#n=60# and #Deltax = (b-a)/n = (pi-0)/60 = pi/60#

#x_0 = 0, x_1=pi/60, x_2 = (2pi)/60, x_3=(3pi)/60, . . . x_(n-1)=((n-1)pi)/60, x_n = pi#

Plug in the numbers and do the arithmetic.
(If permitted, use a computer spreadsheet for all that arithmetic.)

Note
Your answer should be very close to the exact answer:

#int_0^pi sqrt(1-cos^2x) dx = int_0^pi sqrt(sin^2x) dx #

# = int_0^pi abs(sinx) dx #

# = int_0^pi sinx dx#

# = 2#