Question

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?

Answer 1

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`