Question

Mth 229, Calculus Computer Laboratbi The length of a curve defined by the graph of fx) from x-a to x b is givell U definite integral +dx shown below. Find the length of the sine curve from 0 to 2π as we can see that 7.44 is a Using the sum of the lengths of the selected sides of the upper bound for this of the selected sides of the three triangles Using the lengths lower bound for this arc length. three rectangles shown in the sketch is we can see that 10.28 is arc length an (a)Write out the formula for the arc length using f(x) - sin(x). (b)Write a script file to find this arc length. Use a Riemann sum approximation with n= 10,000 rectangles. Save it as int.m. (o Find the are length and check if this answer is between the lower and upper bound given.

Answer 1

a) The general formula to find the arc length is given by

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

The formula for the arc length using f(x) = sin(x) is given by

= $\int_{0}^{2\prod }[cos(x)^{^{2}}+1]dx$

= $\int_{0}^{2\prod }[(cos(2x)+1)/2+1]dx$

= $\frac{1}{2}\int_{0}^{2\prod }[(cos(2x)+1]dx + \int_{0}^{2\prod }1dx$

= $\frac{1}{2}{\int_{0}^{2\prod }[(cos(2x)dx+\frac{1}{2}\int_{0}^{2\prod }1dx} + \int_{0}^{2\prod }1dx$

= $\frac{1}{2}\left [sin(2x)/2) \right ]_{0}^{2\prod }+\frac{1}{2}\left [ x \right ]_{0}^{2\prod }+\left [ x \right ]_{0}^{2\prod }$

= $\frac{1}{4}\left [sin(4\prod )-sin(0) \right ]_+\frac{1}{2}\left [ 2\prod -0 \right ]^{ }+\left [ 2\prod -0 \right ]_{ }$

= $0+\frac{1}{2}\left [ 2\prod \right ]^{ }+ 2\prod$

= $\prod \right ]^{ }+ 2\prod$

= $3\prod$

Arc length =