View from the Top

Okay! You've climbed that mat221 mountain:

• Integration techniques (parts, trig, trig substitution, approximate integration): arc length, and areas of surfaces of revolution (drawing 3-D)

• Parametric equations and curves (in the plane): where we again encountered arc length, and surface areas, as well as the new idea of Bezier curves;

• Polar coordinates (for situations where circles and rays are more important than Descarte's orthogonal lines): lengths and areas;

• Vectors: 3-D coordinates systems (understanding and drawing 2-D projections of them); operations like the dot and cross products, with their corresponding jobs (detecting orthogonal and parallel vectors, respectively); equations of lines and planes using vectors;

• Vector-valued functions and space curves: derivatives and integrals (three times as much work, generally, and much that's generalized from real-valued functions); arc length, which is a simple generalization of the 2-D formula; Frenet frame, normal, binormal and tangent vectors; the new concept of curvature, related to the constant curvature of the ``osculating circle'', which dictates the ``tummy tickler'' acceleration along a space curve; and motion in space: velocity and acceration, which is again a simple generalization and three times as much work as we did when dealing with real-valued functions.

The focus has been on motion in the plane and in space, and applications of integration such as finding the lengths of such motions, or the area swept out by a curve.