cp /math/calclab/MA1023/NumInt_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `NumInt_start.mws`, and go through it
carefully. Then you can start working on the exercises.

Both methods start by dividing the interval into subintervals of equal length by choosing a partition

satisfying

where

is the length of each subinterval. For the trapezoidal rule, the integral over each subinterval is approximated by the area of a trapezoid. This gives the following approximation to the integral

For Simpson's rule, the function is approximated by a parabola over
pairs of subintervals. When the areas under the parabolas are computed
and summed up, the result is the following approximation.

There are many times in math, science, and engineering that coordinate systems other than the familiar one of Cartesian coordinates are convenient. In this lab, we consider one of the most common and useful such systems, that of polar coordinates.

These are three types of well-known graphs in polar coordinates. The
table below will allow you to identify the graphs in the exercises.

Name | Equation |

cardioid | or |

limaçon | or |

rose | or |

- A point in the plane can have more than one representation in polar coordinates. For example, , is the same point as , . In general a point in the plane can have an infinite number of representations in polar coordinates, just by adding multiples of to . Even if you restrict a point in the plane can have several different representations.
- The origin is determined by . The angle can have any value.

- For each of the definite integrals given below, find the exact value of the integral using the
`int`command. Then find the minimum number of subintervals needed to approximate the integral using the trapezoidal rule and then again using Simpson's rule with and error no greater than 0.001. Which method is better and why? - Find all points of intersection for the pair of curves in polar coordinates given by the equations and for .

2004-11-22