Lesson: Arc Length by Integration

Use the secant between and to get a lower bound on the length of the curve between those points. Give your answer to 3 decimal places.

Use the three additional points at to get a better approximation of this length to 3 decimal places.

Calculate the length of the curve exactly, giving your answer to 4 decimal places.

A table of integrals gives the formula

Use this to compute the exact arc length to 5 decimal places.

Estimate the arc length using the 2 line segments in the first figure, giving your answer to 5 decimal places.

Estimate the arc length using the 4 line segments in the second figure, giving your answer to 5 decimal places.

Using Simpson’s rule with 4 subintervals, the first summand is . What is the second summand in estimating the arc length? Give your answer to 4 decimal places.

Using Simpson’s rule with 4 subintervals of width 0.5, what is the estimated arc length to 5 decimal places?

Using Simpson’s rule with 8 subintervals of width 0.25, what is the estimated arc length to 5 decimal places?

Given that the arc length from to is given by , the first thing we need for an arc length parameterization is the inverse . Determine so that .

Hence, give the arc length parameterization , of the curve.

The arc length between each of the points , and is one unit. Give the coordinates of to 3 decimal places.

Find the derivative of .

Find the derivative of .

Hence, find .

Use your results above to calculate, to 5 decimal places, the arc length of the curve between and .