Lecture 27: Indefinite Integrals and Integration by Substitution

1.1 Basic Concepts of Curvature

What is Curvature

• Literal meaning: Using limits to describe the degree of curvature at a certain point of a curve;
• Related to the point where it is located;
• Essence: describes the change in tangent angle between a curve at different points;
• 

#####Definition: Curvature

Description: The absolute value of the ratio of the tangent angle $\Delta\alpha$ of arc $MM^{\prime}$ to the arc length $\Delta s$ is called the average curvature of the arc, denoted as: $\overline{K}=\begin{vmatrix}\frac{\Delta\alpha}{\Delta s}\end{vmatrix}$

When $M^{\prime}$ tends toward M along the curve L, if the limit of the average curvature of arc $\overline{MM^{\prime}}$ exists, this limit is called the curvature of the curve L at point M, denoted as K, that is:

$K=\lim_{M^{'}\rightarrow M}\left|\frac{\Delta\alpha}{\Delta s}\right|\text{ or }K=\lim_{\Delta s\rightarrow0}\left|\frac{\Delta\alpha}{\Delta s}\right|=\left|\frac{d\alpha}{ds}\right|$

is: $K=\frac{|y^{\prime\prime}|}{(1+{y^{\prime}}^2)^{\frac32}}$

Explanation

• Using limits to describe the degree of curvature at a certain point in a curve;
• Curvature: $$K=\frac{|y^{\prime\prime}|}{(1+{y^{\prime}}^2)^{\frac32}}$$
• Radius of curvature: $$R=\frac1K$$

1.2 Common Situations of Curvature

Straight Line

• The curvature of a straight line is 0;
• Because the change in slope angle is 0;

Circle

• The curvature of a circle is: 1 / radius