Lec 31 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by discussing the transition from integral formulas to geometry, which serves as a foundation for multivariable calculus. The primary focus is on the concept of arc length, which is introduced through the analogy of a roadway with mileage markers. The arc length, denoted by 's', is explained as the length along the road. The approximation of arc length is likened to connecting points on the curve with straight lines, which becomes increasingly accurate as the divisions become smaller. The formula for calculating the arc length increment, 'delta s', is derived from the Pythagorean theorem, relating the horizontal and vertical distances between points on the curve. The infinitesimal approach of calculus is emphasized, leading to the formula for arc length, which involves the square root of the sum of the squares of differentials of x and y (dx and dy). The professor also clarifies common notational conventions, such as the use of 'ds' to represent the differential of 's', not the product of 'd' and 's'.

The professor elaborates on the formula for arc length, presenting it in multiple forms to emphasize its versatility. The primary form involves the square root of '1 + (dy/dx)^2 dx', which is derived from the basic differential form. Variants of this formula are introduced, such as factoring out 'dx' to yield '1 + (dy/dx)^2', which is particularly useful for practical applications. The professor also addresses common notational shorthand, such as 'integral ds', which is used informally in many textbooks. Additionally, the professor clarifies misconceptions, such as the incorrect assumption that 'f''(x) = (f'(x))^2', by providing a counterexample. The importance of understanding the limits of integration in relation to the variable 's' used for arc length is also discussed, distinguishing 's0' and 'sn' as mileage markers along the curve, as opposed to 'a' and 'b' which are points on the x-axis.

The professor provides examples to illustrate the calculation of arc length. The first example involves a linear function, 'y = mx', where the arc length is calculated from 0 to 10, resulting in a straightforward application of the Pythagorean theorem. This example serves to demonstrate that while simple, the principles of calculus enable the calculation of arc length for any function, not just linear ones. The second example involves a semicircle defined by 'y = sqrt(1 - x^2)', where the derivative 'y'' is calculated to find the arc length element 'ds'. The integral for the arc length of the semicircle is set up, leading to a more complex calculation that highlights the application of calculus to curved functions.

The professor delves into the geometric interpretation of arc length, using the example of a circle to establish a connection between arc length and radians. By calculating the arc length of a semicircle, the professor derives the relationship between the angle in radians and the arc length, showing that 'sin(alpha) = a', where 'alpha' is the angle in radians and 'a' is the horizontal distance along the circle's arc. This leads to a foundational definition of radians and the sine function, emphasizing that the sine of an angle is defined by the ratio of the opposite side to the hypotenuse in a right-angled triangle. The professor highlights the significance of this approach in providing a rigorous definition of trigonometric functions based on first principles.

The professor transitions from two-dimensional arc length to three-dimensional surface area, introducing the concept as a natural extension of the principles learned in multivariable calculus. The focus is on the surface area of rotation, using the previously discussed parabola 'y = x^2' as an example. The surface area is visualized as the outer 'skin' of a three-dimensional shape formed by rotating the parabola around the x-axis. The formula for surface area involves the circumference of the circles formed at each point of rotation, multiplied by the arc length of the curve. The professor emphasizes the importance of understanding the simplest cases, such as linear functions, as a foundation for tackling more complex problems in three-dimensional space.

The professor discusses the calculation of surface area for a sphere and a paraboloid. For the sphere, the formula for surface area is derived by considering a small section of the sphere's surface and integrating over the entire surface. The result is a simple formula involving the radius of the sphere, '2 pi a^2', which is verified by considering special cases like a hemisphere and the entire sphere. The professor also revisits the paraboloid example, setting up the integral for surface area by considering the rotation of the parabola 'y = x^2' around the x-axis. The integral involves the square root of '1 + 4x^2', which was previously derived for arc length, and emphasizes the method and setup of the integral rather than the computation itself.

The professor concludes the lecture by introducing parametric equations and their significance in multivariable calculus. Parametric equations are presented as a way to describe curves where 'x' and 'y' are both functions of a parameter 't', which can be thought of as time. The professor uses the example of a circle defined by 'x = a cos t' and 'y = a sin t' to illustrate how parametric equations can reveal the shape of a curve and its trajectory over time. By squaring and adding the parametric equations, the professor demonstrates that the curve is a circle with radius 'a'. The direction of the trajectory on the circle is also discussed, indicating that the curve is traversed counterclockwise. The lecture ends with a preview of the next topic: arc length in the context of parametric equations.